percipitation of solid

8/16/2019 Percipitation of Solid

1/13

P r e c i p i ta t io n o f S o l id s in E l e c t r o c h e m i c a l e l ls

Keh-Chang Tsaur and Richard Pollard

Departme nt of Chemical Engineering University of Houston Houston Texas 77004

A B S T R A C T

A mathematic al model is developed for an electrochem ical cell in which a sparingly soluble product of an electro-

chemical reaction is precipitated. This model considers the time dependence and the position depe ndence of the system

behavior and includes the interactions betwe en mul tic ompo nent transport of electrolyte species in porous media and si-

multa neous chemical and elect rochemic al reaction kinetics. The model predicts that cell performa nce is sensitive to the

magni tude of the effective rate constant for precipitat ion as a result of kinetic, thermodyn amic , and morp hological effects.

In addition, the analysis indicates that local mass transport of the soluble reaction produ ct b etwee n the electrode surface

and the bulk electrolyte within a pore can influence cell behavior significantly but that redistributi on of the dissolved spe-

cies across a cell is usually unimportant. The calculations also suggest that a local voltage min imu m can be obtained dur-

ing discharge of a cell, depending on the relative rates of nucleation and growth of the solid product. Quant itative informa-

tion on the preci pita tion o f LiC1 in a Li/SOC12 static cell with a cidic ele ctrol yte is presen ted. As part o f the analysis for this

system, a consistent set of transport equations is develop ed for a concentrate d mi xture of two binary electrolyte s in two

neutral solvents.

M a n y e l e c t r o c h e m i c a l c e l l s h a v e s o l i d r e a c t i o n p r o d -

u c t s , a n d t h e s e c a n s i g n i f i c a n t l y a f f e c t s y s t e m p e r f o r m -

a n c e . I n t h e l e a d - a c i d b a t t e r y , t h e n u c l e a t i o n a n d g r o w t h

o f P b S O 4 c r y s t a l s c a n i n f l u e n c e t h e s h a p e o f t h e d i s -

c h a r g e c u r v e ( 1 , 2 ) . S i m i l a r l y , v o l t a g e m i n i m a i n M n O 2 ,

T i O ~ , a n d T 1 2 0 3 e l e c t r o d e s h a v e b e e n a t t r i b u t e d t o n u c l e -

a t i o n e f f e c t s ( 2 , 3 ) . F o r z i n c e l e c t r o d e s w i t h a l k a l i e l e c t r o -

l y t e s , t h e o r e t i c a l c a l c u l a t i o n s h a v e s u g g e s t e d t h a t t r a n s -

p o r t o f a . s o l u b l e z i n c a t e i o n i n t e r m e d i a t e c a n i n f l u e n c e

t h e d i s t r i b u t i o n o f t h e Z n O p r e c i p i t a t e ( 4 ) . A l s o , i t h a s

b e e n s h o w n t h a t d i s s o l u t i o n - p r e c i p i t a t i o n p h e n o m e n a

a r e i m p o r t a n t i n A g / A g C l a n d C d / C d ( O H ) ~ c o u p l e s a n d

t h a t t h e s e e l e c t r o d e s c a n f a i l b y p l u g g i n g o f p o r e s w i t h

r e a c t i o n p r o d u c t s ( 5 ) . F u r t h e r m o r e , i n t h e l i t h i u m , t h i o -

n y l c h l o r i d e c e l l w i t h L i A I C I 4 , A I C l 3 e l e c t r o l y t e , t h e f r o n t

f a c e o f t h e p o r o u s c a r b o n c a t h o d e b e c o m e s p l u g g e d w i t h

L i C l c r y s t a l s a n d t h i s t e r m i n a t e s t h e d i s c h a r g e ( 6 - 3 ) .

A n a n a l y s i s o f p o r o u s e l e c t r o d e s w i t h s p a r i n g l y s o l u -

b l e r e a c t a n t s h a s i n d i c a t e d t h a t d i s s o l u t i o n a n d d i f f u s i o n

o f a n a c t i v e s p e c i e s s h o u l d b e f a s t e n o u g h t o a v o i d

s i g n i f i c a n t m a s s t r a n s p o r t l i m i t a t i o n s b u t n o t t o o f a s t t o

i n c u r s e l f - d i s c h a r g e p r o b l e m s ( 9 ) . H o w e v e r , f o r c e l l s

w i t h r e a c t i o n p r o d u c t s t h a t a r e d e p o s i t e d f r o m t h e e l e c -

t r o l y t e , t h e d e p e n d e n c e o f c e l l p e r f o r m a n c e o n n u c l e a -

t i o n a n d g r o w t h o f a s o l i d p r o d u c t h a s n o t b e e n d e t e r -

m i n e d . I n t h i s p a p e r , t h e i n f l u e n c e o f t h e s e f a c t o r s o n t h e

q u a n t i t a t i v e b e h a v i o r o f t h e l i t h i u m , t h i o n y l c h l o r i d e p r i -

m a r y c e l l i s u s e d t o i l l u s t r a t e t h e e f f e c t s t h a t c a n b e e x -

p e c t e d i n g e n e r a l . C h a n g e s i n c e l l v o l t a g e a n d l i f e t i m e

a r e e l u c i d a t e d a s w e l l a s t h e i m p o r t a n c e o f o p e r a t i n g

c o n d i t i o n s , s p e c i e s r e d i s t r i b u t i o n , c r y s t a l l i z a t i o n r a t e

c o n s t a n t s , l o c a l t r a n s p o r t p h e n o m e n a , a n d t h e m a g n i -

t u d e o f t h e s o l u b i l i t y p r o d u c t . P h y s i c a l e x p l a n a t i o n s a r e

p r o v i d e d f o r t h e e f f e c t s p r e d i c t e d b y t h e m o d e l . A l s o , t h e

r e s u l t s f o r t h e L i / S O C I 2 s y s t e m a r e r e l a t e d t o t r e n d s i n

b e h a v i o r f o r o t h e r c e l l s t h a t i n v o l v e p r e c i p i t a t i o n o f a

s p a r i n g l y s o l u b l e r e a c t i o n p r o d u c t . I n t h i s w a y , g e n e r a l

c o n c l u s i o n s a r e r e a c h e d c o n c e r n i n g t h e i n f l u e n c e o f p r e -

c i p i t a t i o n r a t e c o n s t a n t s a n d a s s o c i a t e d e l e c t r o l y t e p r o p -

e r t i e s o n c e l l d i s c h a r g e c h a r a c t e r i s t i c s .

T h e t h e o r e t i c a l a n a l y s i s f o r t h e L i / S O C I 2 c e l l i s b a s e d

o n a p r e v i o u s t r e a t m e n t ( 7 , 8 ) w h i c h d i d n o t c o n s i d e r t h e

p r e c i p i t a t i o n p r o c e s s b u t , r a t h e r , a s s u m e d t h a t t h e L i C l

c r y s t a l s w e r e a d i r e c t p r o d u c t o f t h e e l e c t r o c h e m i c a l r e -

a c t i o n . T h e m o d e l p r e s e n t e d h e r e r e g a r d s c h l o r i d e i o n s

a s a r e a c t i o n p r o d u c t , a n d i n c l u d e s t h e p o s s i b i l i t y f o r

t h e s e i o n s t o b e t r a n s p o r t e d a w a y f r o m t h e r e a c t i o n s i t e

i n t h e e l e c t r o l y t e . P r e c i p i t a t i o n o f L i C l i s t r e a t e d a s a

s e p a r a t e s t e p . A d d i t i o n a l p h y s i c a l p h e n o m e n a d e s c r i b e d

by the mod el are mul t ip le e lect rochem ical reactions , s i -

multaneous chemical reactions, changes in porosity and

electrolyte composition, ohmic potential drop and diffu-

sion potential in the electrolyte, ohmic potential drop in

the matr ix , and cel l tempe rature var ia t ions . Var iable

physical properties are used in the calculations wherev er

Electrochemical Socie ty Active Member.

exper i menta l inform at ion is avai lable . As par t of the

analysis , a consistent set of transport equations is devel-

oped for a concentra ted mixture of two binary e lect ro-

lytes in two neutral solvents . These equations can be

used to describe the behav ior of many different electro-

chemical cells .

Model Deve lopment

A o n e - d i m e n s i o n a l m o d e l i s p r e s e n t e d f o r a p r i s m a t i c

l i t h i u m , t h i o n y l c h l o r i d e c e l l . T h e c e l l c o n s i s t s o f a l i t h -

i u m s h e e t n e g a t i v e e l e c t r o d e , a L i C l f i l m a d j a c e n t t o t h e

l i t h i u m , a p o r o u s s e p a r a t o r , a r e s e r v o i r o f e l e c t r o l y t e ,

a n d a p o r o u s c a r b o n c a t h o d e . C u r r e n t c o l l e c t o r s a r e l o -

c a t e d a t t h e b a c k o f e a c h e l e c t r o d e . [ A s c h e m a t i c d i a -

g r a m o f t h e s y s t e m i s p r e s e n t e d i n R e f . ( 7 ) ] . T h e k i n e t i c

r e l a t i o n s h i p s a n d t r a n s p o r t e q u a t i o n s u s e d t o p r e d i c t t h e

d i s c h a r g e b e h a v i o r o f t h e c e l l a r e d e s c r i b e d b e l o w .

Cell rea ctions.--The complica ted processes in the posi-

t ive e lect rode are represented by two s imul taneous

reactions

2A1C13+SOC12+2e-~-2A1CI ~ +1 /2SO2+ VzS [I]

SOC12 + 2e- ~ 2CI + /2 S02 + I/2 S [II]

In the presence of lithium ions, Cl ions produced by re-

action [II] can precipitate to form solid LiC] crystals

Li + + CI ~ LiCl(s) [III]

R e a c t i o n s [ I I ] a n d [ I I I ] r e p l a c e t h e r e a c t i o n

2 L i § + S O C I 2 + 2 e ~ 2 L i C l ( s ) + / 2 S O 2 + V 2 S

c o n s i d e r e d p r e v i o u s l y ( 7 ) , i n w h i c h s o l i d L i C I i s r e g a r d e d

a s a d i r e c t p r o d u c t o f t h e e l e c t r o c h e m i c a l r e a c t i o n .

I n t h e n e g a t i v e e l e c t r o d e , t h e o v e r a l l d i s c h a r g e r e a c -

t i o n i s

L i ~ L i + + e [ I V ]

A p r o t e c t i v e f i l m o v e r t h e l i t h i u m c a n b e f o r m e d b y r e a c -

t i o n [ I I I ] a n d b y a r e a c t i o n b e t w e e n L i a n d t h e s o l v e n t a c -

c o r d i n g t o ( I 0 )

2 L i + S O C I 2 - - > 2 L i C l ( s ) + i /2 S O 2 + / 2 S I V ]

S o l i d l i t h i u m c h l o r i d e a t e i t h e r e l e c t r o d e c a n b e d i s -

s o l v e d c h e m i c a l l y b y t h e a c i d

L i C l ( s ) + A i C l 3 - -> L i + + A I C l 4 - [ V I ]

R a t e e x p r e s s i o n s . 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 s a t

t h e c a t h o d e , t h e B u t l e r - V o l m e r e q u a t i o n

ainj = aioj[ea~

e ~F~JRT]

[1]

is used to relate the local current density i,j due to reac-

tion j to the surfa ce ov erpo ten tia l, ~j = ~ - Uj,w, whe re

the theoreti cal open-circui t potential for reaction j is ex-

pressed as

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2/13

V o l . 1 3 3 , N o . 1 1

P R E C I P I T A T I O N I N E L E C T R O C H E M I C A L C E L L S

2 2 9 7

U j , = U j o _ u , o + R T ~ .~ ( c ~ , , ~ ]

9 n r e ~ Si.re i n

\ Po /

n j F s i .j in [2 ]

\

Po /

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 io~ i n E q . [1 ] i s w r i t t e n a s

io j = io~,~f I I (Ci .wlci.~O TM [3]

i

w h e r e t h e p r o d u c t s i g n r a n g e s o v e r io n i c s p e c ie s i n t h e

e l e c t r o l y t e . T h e e x p o n e n t V ii i s g i v e n b y ~cJ s i , , / n j f o r s u >

0 a n d b y - ~ , j s i . j / n j f o r s u < 0 , w h e r e s u i 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 f o r s p e c i e s i i n a n e l e c t r o d e r e a c t i o n j d e -

s c r i b e d b y

~ s i , j M , z i =

n ~ e -

[4]

E q u a t i o n s [1]-[4] a r e a p p r o p r i a t e f o r m a n y e l e c t r o c h e m i -

c a l s y s t e m s , b u t t h e y a s s u m e i m p l i c i t l y t h a t t h e e le c -

t r o d e r e a c t i o n s a r e r e v e r s ib l e . I f r e a c t i o n s [I] a n d [ II ] a r e

i r r e v e r s i b l e , U ~.w a n d i0~ c o u l d c o r r e s p o n d t o c o r r o s i o n

p o t e n t i a l s a n d c o r r o s i o n c u r r e n t s a n d , a s a r e s u l t , t h e i r

c o m p o s i t i o n d e p e n d e n c e s m a y n e e d t o b e m o d i f ie d . T h e

e f f e c t s o f c h a n g e s i n E q . [2 ] a n d [3] a r e c o n s i d e r e d l a t e r

i n t h e p a p e r . I t is a l s o n o t e d t h a t , f o r i r r e v e r s i b l e r e a c -

t i o n s , T a f e l k i n e t i c s m a y b e m o r e a p p r o p r i a t e t h a n E q .

[1 ]. H o w e v e r , i t h a s b e e n e s t a b l i s h e d p r e v i o u s l y ( 7) t h a t

t h e e x a c t f o r m o f th e p o l a r i z a t i o n r e l a t i o n d o e s n o t

i n f l u e n c e a n y o f t h e p r e d i c t e d t r e n d s i n b e h a v i o r . F o r

t h i s r e a s o n , t h e g e n e r a l r a t e e x p r e s s i o n , E q . [1 ], i s u s e d i n

t h i s s t u d y .

T h e e l e c t r o d e w a l l c o n c e n t r a t i o n ci.w i n E q . [2 ] a n d [3]

i s r e l a t e d t o t h e c o r r e s p o n d i n g c o n c e n t r a t i o n c~ i n t h e

b u l k o f a p o r e b y

~ j S l J

. ~ j F a i n j = a k i ( c l . w - c O [5]

T h i s r e l a t i o n c a n b e r e g a r d e d a s a l o c a l p s e u d o s t e a d y -

s t a te b a l a n c e b e t w e e n m a t e r i a l p r o d u c e d a t t h e e l e c tr o d e

s u r f a c e a n d m a t e r i a l t r a n s p o r t e d f r o m t h e w a l l r e g i o n to

t h e b u l k e l e c t r o l y t e . I n p r i n c i p l e , E q . [5 ] c a n b e w r i t t e n

f o r e a c h s p e c i e s i n t h e e l e c tr o l y t e . H o w e v e r , p r e l i m i n a r y

c o m p u t e r c a l c u l a t i o n s s h o w e d t h a t th e d i f f e r e n c e b e-

t w e e n C~.w a n d c~ c a n o n l y b e i m p o r t a n t i f c~ i s s m a l l .

H e n c e , l o c a l m a s s t r a n s p o r t c a n b e n e g l e c t e d f o r L i + ,

A 1 C 14 -, a n d S O C 1 2. T h e e f f e c t i s a l s o n e g l i g i b l e f o r A 1C I~

b e c a u s e , w h e n C A l C ~ 3b e c o m e s s m a l l , r e a c t i o n [ II ] d o m i -

n a t e s a n d t h e a c i d th a t r e m a i n s d o e s n o t i n f l u e n c e c e l l

b e h a v i o r s i g n i f i c a n t l y ( 8). F o r t h e s e r e a s o n s , i t i s o n l y

n e c e s s a r y t o u s e E q . [ 5] f o r C1 - i o n s . T h e q u a n t i t y a i n E q .

[ 1] a n d [5 ] i s t h e e f f e c t i v e l o c a l s u r f a c e a r e a p e r u n i t

v o l u m e o f t h e e l e c t r o d e . A n e m p i r i c a l e x p r e s s i o n

ELiCI P

i s u s e d t o d e s c r i b e t h e r e d u c t i o n i n t h e i n t e r f a c i a l a r e a

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 s t h a t o c c u r s d u r i n g d i s-

c h a r g e d u e t o p r e c i p i t a t i o n o f L iC 1 o v e r t h e c a t h o d e s u r-

f a ce . T h e m a g n i t u d e o f t h e e x p o n e n t p r e f le c t s t h e m o r -

p h o l o g y o f t h e L i C 1 c r y s t a ls (5). T h i s t r e a t m e n t a s s u m e s

i m p l i c i t l y t h a t c a r b o n s u r f a c e s c o v e r e d b y L iC 1 a r e p a ss i -

v a t e d a n d c a n n o l o n g e r a c t a s s i t es f o r r e a c t i o n s [I ] o r

[ I I ] .

T h e p r e c i p i t a t i o n r a t e i n t h e p o s i t i v e e l e c t r o d e i s g i v e n

b y

r i , i = kT [ (C L i+) (C c l ) K ~ ]

[7]

w h e r e K ~ , i s t h e s o l u b i l i t y p r o d u c t f o r Li C 1 i n t h e e l e c t r o -

l y t e . T h e d r i v i n g f o r c e t e r m i n E q . [7] i m p l i e s t h a t t h e

p r e c i p i t a t i o n p r o c e s s i s k i n e t i c a l ly c o n t r o l l e d , a l t h o u g h

t h i s t e r m w o u l d b e s i m i l a r e v e n i f a n o t h e r p h y s i c a l f a c t o r

w a s d o m i n a n t ( 11 ). T h e p r e c i p i t a t i o n p a r a m e t e r kT i s

e v a l u a t e d u s i n g

k w = kIll N+ ( e L i C l ) m k I I I , G [ 8 ]

w h e r e k l l N a n d k l l G a r e r a t e c o n s t a n t s f o r n u c l e a t i o n a n d

g r o w t h t h a t i n c l u d e t h e e f f e c t s o f m a s s t r a n s f e r f r o m t h e

b u l k s o l u t i o n i n t h e p o r e s t o t h e s u r f a c e o f t h e c r y s t a l s .

A t t h e b e g i n n i n g o f d i s c h a r g e , t h e v o l u m e f r a c t i o n eL~c~o f

L i C1 i n t h e c a t h o d e i s z e r o , a n d i t i s n e c e s s a r y t o n u c l e a t e

c r y s ta l s . S u b s e q u e n t l y , g r o w t h o f e x i s t i n g c r y s t a ls a n d

f o r m a t i o n o f n e w n u c l e i c a n t a k e p l a c e s i m u l t a n e o u s l y .

T h e g r o w t h r a t e i s e x p e c t e d t o b e d i r e c t l y r e l a t e d to t h e

i n t e r f a c i a l a r e a o f c ry s t a l s i n c o n t a c t w i t h t h e e l e c t r o ly t e .

T h i s a r e a i s a s s u m e d t o b e p r o p o r t i o n a l t o (eL iC l)m w h e r e

m i s a m o r p h o l o g i c a l p a r a m e t e r a n a l o g o u s t o t h e e x p o -

n e n t p i n E q . [6]. V a l u e s f o r m a n d p a r e n o t n e c e s s a r i l y

t h e s a m e b e c a u s e t h e i n f l u e n c e o f m o r p h o l o g y o n p re c i p -

i t a t io n a n d o n e l e c t r o c h e m i c a l r e a c t i o n s m a y b e d i ff e r -

e n t . P r e c i p i t a t i o n i n t h e s e p a r a t o r i s n o t c o n s i d e r e d i n

t h e a n a l y s i s , b e c a u s e L i C1 c r y s t a l s h a v e n o t b e e n o b -

s e r v e d i n t h a t l o c a t i o n ( 12 ).

I t is n o t e d t h a t , i n g e n e r a l , E q . [7] s h o u l d u s e s p e c i e s

c o n c e n t r a t i o n s e v a l u a t e d a t t h e c r y s ta l s u r fa c e . H o w -

e v e r , e l i m i n a t i o n o f s u r f a c e c o n c e n t r a t i o n s b y c o n s i d e r -

i n g t r a n s p o r t f r o m t h e b u l k t o t h e s u r f a c e t e n d s t o g i v e

e x p r e s s i o n s f o r rE ,, a n d kT w i t h t h e s a m e f o r m a s E q . [ 7]

a n d [ 8] . C o n s e q u e n t l y , k,I~,N a n d k lll,C a c c o u n t f o r t h e c o m -

b i n e d e f f e c ts o f l o c a l d i f f u s i o n a n d n u c l e a t i o n a n d

g r o w t h p h e n o m e n a .

T h e r a t e o f L iC 1 d i s s o l u t i o n i s a s s u m e d t o b e p r o p o r -

t i o n a l t o t h e l o c a l a c i d c o n c e n t r a t i o n a n d t o t h e i n t e r f a -

c i a l a r e a o f L iC 1 in t h e e l e c t r o l y t e , i . e .

rv[ = kwC,lcls (eUCl) m [9]

A t t h e a n o d e , t h e ra t e e x p r e s s i o n s a r e m o d i f i e d t o

r e f l ec t th e p l a n a r n a t u r e o f t h e e l e c t r o d e a n d t h e p r e s -

e n c e o f a f i l m o f L i C1 o v e r t h e l i t h i u m ( 7, 8 ). T h e p o t e n t i a l

c h a n g e a c r o s s t h e L i / L iC 1 i n t e r f a c e i s d e t e r m i n e d d i -

r e c t l y f r o m E q . [1] b e c a u s e t h e r e i s o n l y o n e e l e c t r o d e r e -

a c t i o n a n d , h e n c e , i ,~ v c a n b e r e p l a c e d w i t h t h e

s u p e r f i c i a l c u r r e n t d e n s i t y I . A l s o , io lv i s a s s u m e d t o b e a

c o n s t a n t . T h e v o l t a g e d r o p a c r o s s t h e L i C 1 f i lm is o b -

t a i n e d f r o m O h m ' s l a w u s i n g a c h a r a c t e r i s t i c f i l m co n -

d u c t i v i t y a n d t h e f i h n t h i c k n e s s Of. C h a n g e s i n t h e t h i c k -

n e s s a r e d e p e n d e n t o n th e r a t es o f c h e m i c a l r e a c t i o n s

[ II I ], IV ] , a n d [ V I i, in a c c o r d a n c e w i t h

d ~ f

- ~ SLiCl.krk'(GlCl [10 ]

d t chemical

reactions,k

T h e r e a c t i o n r a t e r il l' p e r u n i t c r o s s - s e c t i o n a l a r e a o f

t h e e l e c t r o d e i s e x p r e s s e d a s

r iH = kT [ ( C u +)b ( C c l ) b - - K s , ]

[11]

w h e r e b i n d i c a t e s t h e f i l m - s o l u t i o n i n t e r fa c e . T h e s u r f a c e

a r e a a v a i l a b l e f o r g r o w t h o f t h e f i l m is t a k e n t o b e c o n -

s t a n t 9

F o r r e a c t i o n [ V ], i t i s a s s u m e d t h a t t h e r e a c t i o n r a t e a t a

g i v e n t i m e i s e q u a l t o t h e p r e v a i l i n g f l u x o f S O C1 2

t h r o u g h t h e f i lm to t h e l i t h i u m s u r f a c e. I f c o n v e c t i o n o f

S O C1 2 i n t h e f i l m is n e g l e c t e d , a n d i f f i lm p r o p e r t i e s a r e

c o n s t a n t , t h e r a t e e q u a t i o n c a n b e w r i t t e n a s

rv ' = N o' = kv '[(Csocl2)b -- (Csocl2)a] [12]

w h e r e k 'v i s a n e f f e c t i v e m a s s t r a n s f e r c o e f f i c i e n t a n d

s u b s c r i p t a r e p r e s e n t s t h e e l e c t r o d e / f i l m i n t e r f ac e . R e a c -

t i o n [ V ] i s e x p e c t e d t o b e v e r y f a s t a n d , t h e r e f o r e , ( cso cl2 )a

i s t a k e n t o b e z e r o .

T h e r a t e o f r e a c t i o n [ V Ii at t h e a n o d e i s g i v e n b y

rv l ' = kvJ(CAIcU)b [13]

T h i s e x p r e s s i o n i s s i m i l a r t o E q . [9], b u t t h e s u r f a c e a r e a

a t w h i c h d i s s o l u t i o n c a n o c c u r i s a s s u m e d t o b e c o n -

s t a n t .

T r a n s p o r t e q u a t i o n s . - - T h e e q u a t i o n s f o r t h e p o r o u s

c a t h o d e a n d s e p a r a t o r a r e b a s e d o n t h e m a c r o s c o p i c

t h e o r y o f p o r o u s e l e c t r o d e s i n w h i c h t h e s o l u t i o n a n d

m a t r i x p h a s e s a r e t r e a t e d a s s u p e r p o s e d c o n t i n u a ( 13 ).

W i t h t h i s a p p r o a c h , a m a t e r i a l b a l a n c e r e l a t i o n f o r s p e -

c i e s i c a n b e e x p r e s s e d a s

O(ec~)

a j i n

- V " Ni [14]

O t




3/13

98

J E lectrochem Soc :

ELECTROCHEMICAL SCIENCE AND TECHNOLOGY

November I986

w h e r e t h e s u p e r f ic i a l t r a n s f e r r a t e

a j ~ ,

o f s p e c i e s i f r o m

t h e s o l i d p h a s e s t o t h e p o r e s o l u t i o n i s r e l a t e d t o t h e

c h e m i c a l a n d e l e c t r o c h e m i c a l r e a c t i o n r a te s b y (8 )

Si, j

a j , , = - - S , , k r k- - . ~ a i m

[151

k

T h e e l e c t r o l y t e i s r e g a r d e d a s a c o n c e n t r a t e d m i x t u r e

o f L i + i o n s , A t C 14 - i o n s , a n d C 1 - i o n s i n A 1 CI ~ a n d S O C I ~

( s p e c i e s 1 , 2 , 3 , 4 , a n d 0 , r e s p e c t i v e l y ) . C h l o r i d e i o n s a r e

i n c l u d e d i n t h e c o n c e n t r a t e d s o l u t i o n a n a l y s i s f o r t h e

s a k e o f g e n e r a l i t y a n d b e c a u s e Cc, c o u l d b e l a r g e i f t h e

p r e c i p i t a t i o n r a t e is l o w . T h e r e a c t i o n p r o d u c t s S a n d

S O~ a r e n o t c o n s i d e r e d i n t h e e q u a t i o n s b e c a u s e t h e i r

p r e s e n c e i n th e e l e c t r o l y t e d o e s n o t c h a n g e co

s i g n i f i c a n t l y , e v e n a t t h e e n d o f d i s c h a r g e , a n d b e c a u s e

t h e y u s u a l l y d o n o t i n t e r f e re w i t h t h e d i s c h a r g e p r o c e s s

(1 4) . I n v e r s i o n o f t h e m u l t i c o m p o n e n t d i f f u s i o n e q u a -

t i o n s y i e l d s e x p l i c i t r e l a t i o n s h i p s f o r th e f l u x e s

N 2 = - - ~ q y 2 A ( D A A V C A ~ - D A B V C B ~ - D A 4 V C 4 )

~ 2 ~ 2

+ + V~AC~V"

[16]

z e F

N 3 = - - e Q v 3 B D B A V C A + ~Vc~ + D ~ 4 V c 4 )

+

z 3 F

N 4 =

--eq(D4AVCA b

D~ sVc~ + D4 4 Vc4 )

t~'i~

+

N o

= - - e q D o A V C h + D o ~ V c , D o 4 V c 4 )

+ v ~ c ~ v "

[17]

+ c4 v"

[18]

F

toni

+ + CoY [19]

T h e d e r i v a t i o n o f t h e s e e q u a t i o n s i s p r e s e n t e d i n t h e A p -

p e n d i x t o g e t h e r w i t h f u n d a m e n t a l r e l a t i o n s h i p s f o r t h e

t r a n s p o r t p r o p e r t i e s . T h e s y m b o l s A a n d B re f e r to t h e

s a l t s L i A 1C 1 4 a n d L i C 1, r e s p e c t i v e l y . T h e t e r m s v~h a r e

s t o i c h i o m e t r i e s f o r d i s s o c i a t io n o f t h e i n d i v i d u a l s a lt s , as

d e s c r i b e d b y

MA v lAMI ~ v 2 A M 2 z2

[20]

M B = v~ B M I Z l - [ - v a B M 3 z : I

w h e r e t h e c h a r g e n u m b e r z~ i s r e l a t e d t o s t o i c h i o m e t r y

b y ~ z~ v~ = 0 . I t is a l s o n o t e d t h a t a n e x p r e s s i o n f o r N ,

i s n o t g i v e n a b o v e b e c a u s e i t c a n b e o b t a i n e d d i r e c t l y

f r o m

i x = F ~ z i N i [ 2 1 ]

S u b s t i t u t i o n o f E q . [ 16 ]- [1 9] i n t o E q . [ 14 ] a n d r e a r r a n g e -

m e n t u s i n g E q . [ A -2 ] a n d [ A -3 ] g i v e s a n o v e r a l l c o n s e r v a -

t i o n e q u a t i o n

- - + V 9 v " = a

32n

t

- - j 3 , + V 4 j4 . + V o j o .

y 3 B

v - i [ a t ? ]

- - Vo t o

F o 2 A z 2 v 3 B z 3

+ - - + V 4V t4" + V oV to"

F

I] Az

I~3BZ3

- - ~ q V C A D A A V V A +

DuAVVB + D4AVV4 - DoAVVo)

- - ~"VCB" (DABVVA + DBBVVB + D4BV V, - DoBVVo)

-

(qVc4" (DA4VVA + DB4VV, + D,~VV ~ - Do , VVo) [22]

A v o l u m e b a l a n c e

= __-~.~. a i . j -

9 ~ v S~.kV,rk

[23]

phases

f o r t h e s o l i d p h a s e s i s u s e d t o in d i c a t e h o w e l e c t r o d e p o -

r o s i t y c h a n g e s a s a r e s u l t o f c h e m i c a l a n d e l e c t r o c h e m i -

c a l r e a c t i o n r a t e s . C o m b i n a t i o n o f E q . [ 22 ] a n d [ 23 ], w i t h

t h e a s s u m p t i o n o f c o n s t a n t p a r t i a l m o l a r v o l u m e s a n d

t r a n s f e r e n c e n u m b e r s , g i v e s a n e x p l i c i t r e l a t i o n s h i p f o r

t h e r e f e r e n c e v e l o c i t y

F v 2 A z 2

+ ~ ' B t 3 J

- + + V o t o

y 3 B z 3

a i n j - - - - , rh

k

phases

T h e c u r r e n t d e n s i t y i2 i n t h e s o l u t i o n p h a s e t h a t a p -

p e a r s i n E q . [ 24 ] i s g i v e n b y

V . i~_ = Eja i , j [251

T h i s e q u a t i o n s t a te s t h a t t h e t r a n s f e r c u r r e n t p e r u n i t

e l e c t r o d e v o l u m e i s e q u i v a l e n t t o t h e s u m o f t h e e l e c tr o -

c h e m i c a l r e a c t i o n r a te s .

T h e m o v e m e n t o f e l e c t r o n s i n t h e m a t r i x p h a s e i s g o v -

e r n e d b y O h m ' s l a w

i, = -crY(b1 [26]

w h e r e r i s t h e e f f e c t i v e m a t r i x c o n d u c t i v i t y g i v e n b y u~ (1

- e ) Q. T h e r e l a t i o n s h i p b e t w e e n t h e s o l u t i o n p o t e n t i a l (P~

a n d t h e c u r r e n t d e n s i t y i n t h e s o lu t i o n p h a s e i s d e r i v e d

b y i n v e r s i o n o f t h e m u l t i c o m p o n e n t d i f f u s i o n e q u a t i o n

( 15 ) . F o r t h e L iA 1 C 14 , L i C 1 , A1 C1 3, S O C 1 2 e l e c t r o l y t e , o n e

o b t a i n s a m o d i f i e d f o r m o f O h m ' s t a w

i2

- - V ~ P 2 - W [ 2 7 ]

K ~ Eq

w h e r e

W = - - - -

A R T s 2 + c A

VcA

C A n F Z 2P 2 A C o

vB RT ( s3 n t~ ~ so )

CBn F ~ + CB VC~

Z31~3B Co

R T

+ c 4 n F s 4 + n t 4 " - - - c 4 V C 4

[28]

Co

p r o v i d e d t h a t a c t i v i t y c o e f f i c ie n t s a r e t a k e n t o b e c o n -

s t a n t . T h e p o t e n t i a l i n E q . [ 27 ] i s m e a s u r e d r e l a t i v e t o a

r e f e r e n c e e l e c t r o d e a t w h i c h n e l e c t r o n s a r e t r a n s f e r r e d

a n d st i 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 fo r s p e c i e s i. C o m -

b i n a t i o n o f E q . [ 26 ], E q . [ 27 ], a n d t h e c o n d i t i o n i~ + i2 = I ,

y i e l d s an e x p r e s s i o n

V ~ = - - + i 2 - - - + W [29 ]

K~E q (]r

f o r t h e v a r i a t i o n o f t h e p o t e n t i a l d i f f e r e n c e , ~ = q ), - q )2 ,

b e t w e e n t h e m a t r i x p h a s e a n d t h e e l e c t ro l y t e .

C h a n g e s i n c e ll t e m p e r a t u r e a r e d e t e r m i n e d f r o m

d T ( U o j - OU~ I v i e

~ = [ ~

T o f f p ( - a i , j )

- - ] ~ d y -

- h o ( T - T n ) + h H a . , , i ( r , . + ~ P r m d y )

I o )

A H R . v r v + A H R . v [

rv , ' +

r v , d y

[30]




4/13

V o l . 1 3 3 , N o . 1 1

P RECI P ITA TI O N IN ELECT RO CH E M ICA L CELLS 2299

This equa tion is based on the f irs t law of therm odynam -

ics and includes the heat generated by chemical reac-

tions, as well as reversible heat effects , Joule heating,

overpotent i a ls associa ted with e lect rode react ions , and

heat transfer to the en vironme nt. The total voltage in Eq.

[30] is obtained by integrati on of matri x and ele ctrol yte

poten tial gradi ents across the cell (16).

The li thium, thionyl chloride system can be described

by 10 varia bles and their gov erni ng equations. For an

electro ly te wi th f ive species , three concentra t ion s (say,

c~, c3, and c4) are independent; the other two composi-

tions can be eliminated using the relation ~ c ~ V ~ = 1

and the constraint of electroneutrality, ~ z ~ c ~ = 0. Con-

sequently, the system variables are i , i, i n n C3.w, C2, C3, C4, r

V~ i2, and ~ and the corresponding equations are two Eq.

[1], Eq. [5], three Eq. [14], Eq. [23], Eq. [24], Eq. [25], and

Eq. [29]. The boundary conditions and initial conditions

for these eq uat ions have been descr ib ed in deta i l previ-

ously (7), and th ey are not re peat ed here. Th e set of equa-

tions are cast into finite diffe rence form accu rate to O(h ~)

and are solved simultaneously by a numerical technique

(17). Each t ime- depen dent e quat ion is programm ed be-

tw een t h e o l d t i m e a n d t h e p r e s e n t t i m e w i t h a r a t i o o f

o l d : n e w = 2 : 3 i n o r d e r t o a t t a i n s t a b i l i t y

R e s u l t s a n d D i s c u s s i o n

Concentra t ion prof i les for chlor ide ions a t several

s tages of a galvanostatic discharge are presented in Fig.

1. Initially, only reaction [I] occurs at the cathode and

chlo ride ions are not pr oduce d. At 50s, reactio n [II] has

just begun, and the nonu niform distr ibution of this reac-

t ion g ives a maxim um in the comp os i t ion prof i le a t the

front face of the electrode. Su bseq uent profiles , and the

overall levels of chloride ions, are sensitive to the rela-

tive rates of C1- ion pro duct ion (reaction [II]) and con-

sum pti on (reaction [III]). The values of the rate constant s

km.N and kIH.G n Eq. [8] are p art ic ula rl y im po rt an t. In th e

exa mpl e shown, km.N is relative ly small, and, there fore, a

large driving force is needed before precipitation can be-

gin. For this reason, the supersaturation level r ises sub-

s tant ia l ly before the precip i ta t ion ra te become s s ignif i -

cant. The small time period (from 50 to 52s) required for

the concentration buildup emphasizes that large quanti-

ties of C1 ions are generat ed q uickl y by reacti on [II] at

the operating current density.

As the a mount of LiC1 deposited locally increases, the

second term in the expression (Eq. [8]) for the effective

precip i ta t ion ra te cons tant becomes more impor tant .

With the larg e val ue for km,G co nsi der ed in Fig. 1, the

ma gni tud e of kT rises rapi dly and, he nce, more LiC1 is de-

posit ed and Cc~- is reduced. Appr oxi mat ely 25s after the

ini tia tion of rea cti on [II] (t ~ 75s), a bal anc e is estab-

l ished betwe en the local rates of product i on and con-

IO-O

k II I N= lOS cm3]m~ - ~ 52

s e c

10 7 kEI,G = 1012 crn3/mol,sec / ~ -- _

l o - s ~6 , ec

o 1 0 - 9 / / ~ 100,eo

50

s e c

10-1o

I

SEPARATOR I POSITIVE ELECTRODE

1 o - c o o 1 0 2 I

o ~ o ~

Distance (mm)

F ig . 1 . P o s it io n d e p e n d e n c e o f c h l o r i d e i o n c o n c e n t r a t i o n a t d i f fe r e n t

t i m e s d u r i n g a c e l l d i s c h a r g e . P h y s i c a l p a r a m e t e r s u s e d i n t h e c a l c u l a -

t i o n s a r e g i v e n i n T a b l e I a n d c e l l s p e c i f i c a t i o n s a r e l i s t ed i n T a b l e I I .

I n th i s d i a g r a m , l o c a l m a s s t r a n s p o r t o f c h l o r i d e i o n s w i t h i n t h e p o r e s i s

n e g l e c t e d ,

i . e . k c ~ -

= o~ an d Cc~- , w = Cc~- .

i

012

sumption and the chlor ide ion concentra t ion no longer

changes s ignificantly with time.

Figure i also shows that the chloride ion concent ration

at the negative electrode/separ ator interface remains at a

relatively low value during the discharge. This can be at-

tr ibuted to moderately fast precipitation at this location.

Fur thermore, the d iagram shows that a large dr iv ing

force for crystallization can exist within the separator.

Therefore, the absence of crystals in this region in practi-

cal cells (12) suggests that the characteris tic nucleation

rate constant for the separator is very small.

The tr ends in beha vior with other value s of k~..N and

kn~ .G are similar to those shown in Fig. 1, but the concen-

trati on chang es are not always so dramatic. With klll.N and

k m , c bot h equ al to 106 cm3/mol 9s, the concentrat ions for t

< 52s are the same as those in Fig. 1. Subs eque ntl y, how-

ever, coL- increas es s lowly until the end of discharge.

Wit h a km.G va lu e of 1015 cm3/m ol 9 and km.N eq ua l t o 106

cma/mol 9 , only a very small rise in chlo ride ion le vel at

the be ginn ing of reaction [II] is obse rved becau se kv

quic kly bec ome s large. In this case, a low degree of su-

persaturation is maintained throughout the discharge.

Dif ferent levels of chlor ide ion concen tra t ion (caused

by d if ferent values for the precip i ta t ion ra te cons tants )

can alter the shape of the disc harge curve. Figure 2

shows predi cted results for four different sets of rate

constants . In each case, the hump at the beginning of dis-

charge corresponds to reaction [I] , which is unaffected

by chloride ion concentration. I t should be noted that the

initial voltage delay shown in Fig. 2 is larger than delays

pred ict ed previ ously (8). The cause of the delay is the

same in both cases but, for the cell in Fig. 2, the electro-

ly te concentra t ion is smal ler , leading to a greater in i t ia l

increase in K and, hence, a larger initial reduction in the

cell resis tance.

The tr ia ngle in Fig. 2 indic ates the t ime at which the

potential difference driving force at the front of the cath-

ode is large enough to initiate reaction [II] . After this

time, the behavior is depen dent on the precipitation rate.

The rapid changes in cel l vol tage that are predic te d un-

der some conditions (see curves b, c, and d) can be di-

rectly related to the concentration changes il lustrated in

Fig. 1. This is because the chloride ion concentration ap-

pears in the express ion for the theoret ical open-circ ui t

potential (see Eq. [2])

R T

U l 1. ~ = U , e - U I v

F

In [ C Li+ . , , ,) CcI -w) / p o~Csoc l2 .w) ~ /2] [31]

The voltage mi nim um for curve b at 52s corresponds ex-

actly to the maximum chloride ion level (see Fig. 1) . In

v

>

(9

5

I--

3 . 0

2 . 5

2 o L -

o

k i l t , G (cm3/mOl sec)

a : 1015

b : 1012

c

109

d : 1 0 5

k~i,N = 105 crna/rnol,sec )

d

i i i

1 2 3 4 5

t ( m i n)

F i g . 2 . T h e o r e t i c a l d i s c h a r g e c u r v es f o r f o u r v a l u e s o f k m . G . P a r a m e -

t e r s a r e a s in F i g . 1 . S y m b o l V i n d i c a t e s t h e i n i t i a t i o n o f r e a c t i o n [ l l ] .

N o t e t h a t , f o r t h e d i s c h a r g e r a t e c o n s i d e r e d h e r e , a t i m e o f 4 m i n c o r r e -

s p o n d s t o a t h i o n y l c h l o r id e u t i l iz a t i o n o f 1 3 . 9 .




5/13

23 J E lectrochem Soc : E L E C T R O C H E M I C A L S C I E N C E A N D T E C H N O L O G Y

Novemb er t986

case d, the voltage drops substantially, but there is no

vol tage minimum because the concentra t ion of chlor ide

ions cont inues to r ise throughout the d ischarge. Local

voltage minima have been observed in other cells under

certain operating conditions. Examp les include the lead-

aci d s yst em (1, 2) and el ec tr od es o f MnO2, TiOz, an d T1203

(2, 3). The results obtained here suggest that the minima

may be due to changes in supersaturation level associa-

ted with relatively s low nucleation rates k i l l N

k i n , c ) .

For the Li/SOC12 cell described above, changes in C1-

ion con cent rati on level after t = 75s are too small to alter

the cell voltage apprecia bly; at this stage, ohmic losses in

the separat or and in the LiC1 film at the anode c ontrol the

s lopes of the d ischarge curves. The sharp re duct ion in

volta ge that causes cell failure is due to plugging o f the

pores at the front face of the pos itive elec trode wit h LiC1

crystals (7).

The resul ts in Fig. 2 indica te that the cell l ifetim e is

sensit ive to the mag nit ude of km.G. Inte rpre tat ion of this

effect requires detailed information on the importa nce of

the chlor ide ion conce ntra t ion level in the sys tem. The

chloride ion is an intermedi ate species; i ts concentrati on

influen ces and is influen ced by the rates of reactions [II]

and [III] . In general, the concentration is a function of

both position across the cell and time. However, the su-

persaturatio n level is predicted to be relatively constant

across the positive electrode, as i l lustrated in Fig. 1.

Also, this level does not change app reciably shortly after

react ion [II] begins (see above). Therefore, the overall ef-

fect of km,G on the conc ent rati on l evel can be re pres ent ed

by cc~- at one pa rti cul ar lo cat ion an d ti me (see Fig. 3).

With hi gh km.G e . g . , case a from Fig. 2), the l evel of super-

saturation is very low, and an asymptote corresponding

to fast preci pitat ion is obtained. As km.a is reduc ed, the

chloride ion level in the elect rolyte increases markedl y.

Wh en kil,.G is le ss th an km.N e . g . , case d), the concentra-

tion is dictated by

kin.N;

the larger kt..N, the greater the

quant i ty of LiC1 precip i ta ted by nucleat ion and, hence,

the lower the chloride ion level.

A

~ 1 0 - 5

0

r

L L

0

0

0

0

LU

N 1 0 z

t-

O

-I-...

f-

O

c-

O

o

e--

~ 10 -9

O

O

e-

t = 1 0 0 s e c

,

06cm3 /mo l~

fl

1 0 1 1 ~ i p

1 0 5 1 0 8

1

0 1 1

1

0 1 4

k i n , G c r n 3 / m o l - s e c )

F i g . 3 . D e p e n d e n c e o f c h lo r i d e i o n c o n c e n t r a t io n a t t h e r e s e r v o ir /

p o s i t i v e e l e c t r o d e i n t e r f a c e a t t = 1 0 0 s o n k m N a n d k m G . P a r a m e t e r s

a r e a s i n F i g . 1 . S y m b o l s a b c a n d d re p r e s e n t t h e c a s e s s h o w n i n

F i g . 2 .

The re la t ively h igh conce ntra t ions on the lef t -hand

side of Fig. 3 sugges t that t here m ight be s ignificant re-

distr ibuti on of chloride ions across the cell . Howeve r,

even wi th c once ntra tion s as high as 10 2M, this is not the

case because: (i) the hold-up of chloride ions in the elec-

trolyte is always small com pared to the quant ity of LiC1

deposited e . g . , wi th cc~- ~ 5 x 10-3M, 99 of th e C1- ions

produced are precipitated and with

c c ~ -

= 3 x 10-2M, 97

are precipitated), and i t ) compos i t ion gradient dr iv ing

forces are too small for s ignificant diffus ion of CI- ions

through the e lect rode or separator

e . g . ,

with

C c , - = 3 •

10-2M, less than 2 of the C1- ions pro du ce d at any loca-

tion diffuse away from the local reaction sites). Theoreti-

cal discharg es were re peat ed w ith DBB incre ased by an

order of magn itude but, even at the highest C1 ion con-

centrations, redi str ibution was s ti l l negligible.

Redis t r ibut ion is a lso expected to be unimpor tant in

other cells with solid reaction products because s imilari-

t ies in the so lubi l i ty products e . g . , for PbSO4, Ni(OH)2,

Cd(OH)~, and Fe(OH)=) (1, 30) indicate that concentra-

tions, and hence co ncentration gradients , will be s imilar

to those pred icte d for the Li/SOC12 system. In contrast, if

ver y high co nce ntr at ion s (O(10-1M)) of zinc ate ions are

obtained in the zinc electrode, substantial redistr ibution

may occur (4).

In the abse nce of redis t r ibut ion , ch lor ide ions pro-

duced locally by reaction [II] are precipitated in the same

location by reaction [III] . Therefore, the dis tr ibutions of

the e lect roch emical react ion and the so l id LiC1 volum e

fraction correspond directly. I t has been esta blished pre-

viously (7) that cell lifetime for the Li/SOC12 system is di-

rectly re lated to the di s tr ib utio n of LiC1 crystals ; the

more uniform the dis tr ibution, the longer the cell can op-

erate before plugging of the pores occurs at the front face

of the e lect rode. Therefore , t c e u i s determined pr imar i ly

by the dis tr ibution of electrochel~ical reaction [II]. This

dis t r ibut ion is inf luenced by k i n N and k i n , G (see Fig. 4)

since these parameters alter the chloride ion concentra-

t ion . The in teract ions between concentra t ion and the

rate of reaction [II] are desc ribe d below.

1. The e ffectiv e exc han ge curre nt densit y ioH depe nds

on chloride ion concentra tion in ac cordance with Eq. [3]

/oil

i o l l ,r e f C c l - , w / C c l - , r e f ) a c l l C s o c l 2 , w / C S o c l 2 , r e f ) a a l l/ 2

[32]

E

v

o

2

1 0 5

k i n , N = 1 0 9

c m 3 / m o l . s e c

J

b a .

10 8 101 1 10

k m , G ( c m 3 / m o l ' s e c )

F i g . 4 . D e p e n d e n c e o f c e l l l if e t i m e o n

km N

n d k n [ G . P a r a m e t e r s a r e

a s in F i g . 1 . S y m b o l s a b c a n d d r e p r e s e n t t h e c a s e s sh o w n i n F ig . 2 .




6/13

V o L I 3 3 , N o . 1 I PRECIPITATION IN

A n i n c r e a s e i n c o n c e n t r a t i o n r a i s e s ioH ; t h i s t e n d s t o r e -

d u c e t h e p e n e t r a t i o n d e p t h f o r r e a c t i o n [ II ] (7 , 2 9) a n d t o

d e c r e a s e t h e c e l l l i fe t i m e .

2 . T h e d r i v i n g f o r c e f o r r e a c t i o n [ II ], ~ t = ~ - U ,, w, i s

i n f l u e n c e d b y t h e d e p e n d e n c e o f U H.~ o n t h e c h l o r i d e i o n

c o n c e n t r a t i o n ( s e e E q . [ 3 1] ). F o r t h e L i / SO C 1 2 s y s t e m ,

w h e r e t h e C 1 i o n i s 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 p r o d u c t ,

t h e n e t e f f e c t o f a n i n c r e a s e i n c o n c e n t r a t i o n i s t h a t ~ H

b e c o m e s l e ss n e g a t iv e , w h i c h t e n d s t o m a k e t h e r e a c t io n

d i s t r i b u t i o n m o r e u n i f o r m . H o w e v e r , i t s h o u l d b e n o t e d

t h a t a c h a n g e i n Um w g e n e r a l l y g i v e s a s m a l l e r c h a n g e i n

~ H b e c a u s e -q a l t e r s in a m a n n e r w h i c h t e n d s t o c o u n t e r -

a c t t h e e f f e c t o f U . , w .

3 . T h e d i s t r i b u t i o n o f r e a c t i o n [ I I] a t a n y t i m e i s a f u n c -

t i o n o f a l l p r e v i o u s C1 i o n d i s t r i b u t i o n s a n d c o n c e n t r a -

t i o n s . F o r e x a m p l e , t h e e x p o n e n t m i n E q . [8 ] i n f l u e n c e s

t h e r o l e o f l o c a l v o l u m e f r a c t i o n o f L iC 1 a n d k m,G o n t h e

e f f e c t i v e p r e c i p i t a t i o n r a t e c o n s t a n t k v, a n d t h i s i n t e r a c t s

w i t h r e a c t i o n r a t e t h r o u g h t h e i n t e r f a c i a l a r e a a, g i v e n b y

E q . [ 6 ] .

F a c t o r s 1 -3 c a n o p p o s e o n e a n o t h e r a n d h e n c e g i v e r i s e

t o a m i n i m u m t ~ H, a s s h o w n i n F i g . 4 . W i t h k HI,G = 1 0 1~

c m 3 / m o l 9

(e .g . ,

c a s e a f r o m F i g , 2 ), t h e c e l l l i f e t i m e i s d e -

t e r m i n e d b y t h e d i s t r i b u t i o n o f r e a c t i o n [ II ] i n t h e l i m i t

o f n e g l i g i b l e s u p e r s a t u r a t i o n o f t h e e l e c t r o l y t e ( i .e . , f a s t

p r e c i p i t a t i o n ) . I n t h i s a s y m p t o t e , t h e m a g n i t u d e o f k ~,N

i s n e g l i g i b l e i n c o m p a r i s o n t o t h e k m,~ t e r m i n E q . [ 8] a n d ,

h e n c e , t h e r e s u l t s f o r t h e t w o v a l u e s o f k m,~ a r e e s s e n -

t i a l l y t h e s a m e . W h e n k i . ,c i s r e d u c e d w i t h c o n s t a n t km .N

(e .g . , c a s e s a , b , a n d c ), t h e l i f e t i m e i s r e d u c e d b y a s m u c h

a s a f a c t o r o f t w o . T h i s c a n b e a t t r i b u t e d t o t h e i n c r e a s e

i n io n ( s e e a b o v e ) . I f km .G i s r e d u c e d f u r t h e r

(e .g . ,

s e e c a s e s

c a n d d ) , c h a n g e s i n U ,,w b e c o m e m o r e i m p o r t a n t t h a n

c h a n g e s i n i o, a n d t h e c e l l l i f e t i m e i n c r e a s e s . O n t h e l e ft -

h a n d s i d e o f F i g . 4 , k ~.,N i s g r e a t e r t h a n k ~[,~ , a n d t h e c h l o -

r i d e io n c o n c e n t r a t i o n i s d e t e r m i n e d p r i m a r i l y b y th e

m a g n i t u d e o f kin ,N ; w i t h a l a r g e r k in .N , c c~ - i s l o w e r , a n d

t h e l i f e t i m e i s s l i g h t l y g r e a t e r .

T h e r e s u l t s p r e s e n t e d i n F i g . 1 -4 c o n s i d e r c h l o r i d e i o n

c o n c e n t r a t i o n v a r i a t i o n s a c r o s s t h e c e l l b u t d o n o t in -

c l u d e l o c a l m a s s t r a n s p o r t o f C 1 - i o n s f r o m t h e e l e c t r o d e

s u r f a c e , w h e r e t h e y a r e p r o d u c e d b y r e a c t i o n [ I I] , t o t h e

b u l k o f a n i n d i v i d u a l p o r e . W i t h f i n i t e v a l u e s f o r kc~ , E q .

[5 ] c a n b e i n c l u d e d i n t h e a n a l y s i s t o p r e d i c t t h e d i f f e r-

e n c e b e t w e e n c c~ - a n d cc~-,w . F i g u r e 5 c o m p a r e s t y p i c a l

v a l u e s o f t h e c o n c e n t r a t i o n s o b t a i n e d w i t h d i f f e re n t

v a l u e s o f k .~,N a n d k Hi,a. T h e b u l k c o n c e n t r a t i o n s a r e e s -

s e n t i a l l y u n c h a n g e d f r o m t h e r e s u l t s p r e d i c t e d i n t h e ab -

s e n c e o f l o c a l m a s s t r a n s p o r t ( s ee F ig . 3 ); t h e s u p e r s a t u -

r a t i o n l e v e l t e n d s t o z e ro a s kf~,c i s i n c r e a s e d . W i t h kH~,G =

1 0 ~'~ c m ~ / m o 1 9 S , C ct-.w m u s t b e s e v e r a l o r d e r s o f m a g n i -

t u d e l a r g e r t h a n cc~- i n o r d e r t o s a t i s f y E q . [ 5] . T h e v a l u e

o f C c~ -.w r i s e s w h e n

Cc,-

a n d C c~-.w b e c o m e c o m p a r a b l e i n

m a g n i t u d e . T h i s is n e c e s s a r y t o m a i n t a i n t h e c o r r e c t c on -

c e n t r a t i o n d i f f e r e n c e d r i v i n g f o r c e f o r l o c a l m a s s t r a n s -

p o r t , a s d e f i n e d b y t h e r i g h t - h a n d s i d e o f E q . [ 5] . A s i m i -

l a r d i f f e r e n c e b e t w e e n

Cc~-

a n d c c , .w s t i l l e x i s t s f o r k , ,, ,G <

1 0 s c m ~ / m o l - s e v e n t h o u g h i t is t o o s m a l l t o b e d i s c e r n i -

b l e f r o m t h e d i a g r a m . F o r k m.~ = 1 0 " c m ~ / m o l " s , t h e b u l k

c o n c e n t r a t i o n i s a l w a y s s m a l l b y c o m p a r i s o n w i t h

cc~ ,w

a n d , h e n c e , t h e w a l l c o n c e n t r a t i o n i s a l m o s t i n d e p e n d e n t

o f

k i n c .

T h e m a g n i t u d e o f t h e w a l l c o n c e n t r a t i o n i s i n f l u e n c e d

b y t h e r a t e o f r e a c t i o n [ II ] t h r o u g h E q . [ 5] , b u t i t a l s o

i n f l u e n c e s a i ,H t h r o u g h t h e d e p e n d e n c e o f ioH a n d U H,w o n

cc~-,w a s g i v e n b y E q . [ 31 ] a n d [ 32 ]. C o n s e q u e n t l y , t h e d i s -

t r i b u t i o n o f r e a c t i o n [ II ], a n d , h e n c e , t h e c e l l l i f e t i m e , i s

d e p e n d e n t o n t h e l o c a l v a l u e o f cc~-,~ .

I n F i g . 6 , l i f e t i m e s o b t a i n e d w i t h a f i n i t e v a l u e f o r k c~-

( i .e . , cc , - ~

c c, - ,~ ) a r e c o m p a r e d w i t h r e s u l t s f r o m F i g . 4

( k c i - = co a n d C c t - = C ci , w ). W i t h

k u l . N =

1 0 ~ c m a / m o l " s , t h e

l i f e t i m e i s a l m o s t i n d e p e n d e n t o f k m,~ b e c a u s e c h a n g e s

i n t h e w a l l c o n c e n t r a t i o n a r e v e r y s m a l l . F o r k m.N = 1 0 ~

c m a / m o l 9 a n d l a r g e k l , .~ , t h e w a l l c o n c e n t r a t i o n i s t h e

s a m e a s f o r k l,l,~ = 1 0 " c m a / m o l " s , t h e s a m e l i f e t i m e i s p r e -

d i c t e d . I n t h i s a s y m p t o t e , t h e p r e c i p i t a t io n r a t e c o n -

s t a n t s ar e u n i m p o r t a n t a n d tr i s d e t e r m i n e d b y t h e b a l -

a n c e o f k i n e t i c , o h m i c , a n d m a s s t r a n s p o r t l i m i t a t i o n s

t h a t s e t t h e d i s t r i b u t i o n o f r e a c t i o n [ II ].

E L E C T R O C H E M I C A L C E L L S

I

t = 1 0 0 s e c

2 3 0 1

I

- ' - 0 _ 5 1

1 -

~ ~ = 1 0 6 c m 3 / m o l . s e c

9 k r n , N

- .q

U 1 0 - 7

.2

E

e.-

0 km, N = 1 0 9 c m a / m o l ' s e c

o

~ -

1 0 - 9 _

_o

0 ~ Bu lk

C o n c e n t r a t i o n

r

0 - - ~ W a l l C o n c e n t r a t i o n

1 0 - 1 1 1 I I I ]

1 0 5 1 0 8 1 0 1 1 1 0 1 4 '

k i n , e ( c m S / m o l . s e c )

F i g . 5 . D e p e n d e n c e o f c h l o r id e i o n c o n c e n t r a t i o n a t t h e r e s e r v o i r/

p o s i t i v e e l e c t r o d e i n t e r f a c e a t t = 1 0 0 s o n k m .N a n d k ~ . ~ . P a r a m e t e r s

a r e a s i n F ig . 1 e x c e p t k c l = 2 . 0 c m / s.

I I I

/

k r n N = 1 0 9 c m 3 / m o l . s e c /

/

~ . ~ / k l~ ,N = 1 0 6 c m 3 /m o l . se c

Y

~ 4

3

2 I I

1 0 5 1 0 a 1 0 1 ~ 1 0 1 4

k ~ . G ( c m a / m o l s e c )

F i g . 6 . D e p e n d e n c e o f c e l l li f e t i m e o n k . l. N a n d km , o . P a r a m e t e r s a r e

a s in F i g . 1 e x c e p t k c t - = 2 . 0 c m / s f o r t h e d a s h e d c u r v es , T h e f u l l

c u r v e s a r e f r o m F i g . 4 f o r c o m p a r i s o n .




7/13

2 3 0 2 J . E l e c t r o c h e m . S o c . : E L E C T R O C H E M I C A L S C I E N C E A N D T E C H N O L O G Y N o v e m b e r 1 9 8 6

i

lO-6

1 0 7

1 0 8

1 0 9

1 0 l o

"

~ .

\

k r t l N = 1 0 6 c m3 / mo l . s ec

k T],G= 1012 c m 3 / r n o l . s e c

k c i - = 2 . 0 c m / s e c

0 . 0 0 ' . 1

- - B u lk C o n c e n t ra h o n

- - - - W a l l C o n c e n t r a l i o n

0 1 2 0 . 3

D i s t a n c e t o t h e F r o n t o f P o s i t i v e E l e c t ro d e ( m m )

F i g . 7 . P o s i t i o n d e p e n d e n c e o f c h l o r i d e i o n c o n c e n t r a t i o n a t d i f f e r e n t

t i m e s d u r i n g a c e l l d i s c h a r g e . P a r a m e t e r s a r e a s i n F ig . 5 . t~ - S 2 s , t 2 -

5 6 s , t ~ ~ 1 0 0 s .

T h e d i f f e re n c e i n tcell v a l u e s f o r kCl- = o o a n d kcl- = 2 . 0

c m / s i n t h e h i g h k ,m ~ a s y m p t o t e c a n b e a t t r i b u t e d t o a

c h a n g e i n t h e 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 . T h e r a t e o f r e-

a c t i o n [ II ] i s f a s t e s t a t t h e f r o n t f a c e o f t h e e l e c t r o d e a n d ,

w i t h a f i n i t e v a l u e f o r kc ,- , t h e d i f f e r e n c e b e t w e e n t h e

c o n c e n t r a t i o n s a t t h e w a l l a n d i n t h e b u l k a r e g r e a t e s t a t

t h i s l o c a t i o n ( s e e F i g . 7) . A s a r e s u l t , U~,.~ o b t a i n e d u s i n g

E q . [ 31 ] i s s m a l l e r a t t h e f r o n t f a c e t h a n a t t h e b a c k o f t h e

e l e c t r o d e . T h e p o t e n t i a l d i f f e r e n c e ~ i s a l s o s m a l l e s t a t

t h e f r o n t f ac e , a n d i t s d i s t r i b u t i o n t h r o u g h t h e e l e c t r o d e

i s s i m i l a r t o t h e o n e f o r U n .~ . H e n c e , t h e d r i v i n g f o r c e ~ , ,

f o r r e a c t i o n [ II ] i s r e la t i v e l y u n i f o r m , a n d t h e c e ll l i fe t i m e

i s l o n g e r . I n c o n t r a s t , t h e U ,,.~ d i s t r i b u t i o n w i t h k c l - = oo

i s m o r e u n i f o r m , s i n c e c c l- .~ ( = C c l - ) i s m o r e u n i f o r m ( s ee

F i g . 1 ), w h e r e a s t h e d i s t r i b u t i o n o f ~ i s s i m i l a r . C o n s e -

q u e n t l y , t h e s u r f a c e o v e r p o t e n t i a l d i s t r i b u t i o n i s l e s s

u n i f o r m w i t h

k c , - = ~ ,

a n d t h e c o r r e s p o n d i n g c e l l l i f e -

t i m e i s s h o r t e r .

I n F i g . 6 , a r e d u c t i o n i n k m . c r a i s e s c c,-.w ( s e e F i g . 5 ) a n d ,

h e n c e , r a i s e s i o,,. T h i s e f f e c t , i n c o n j u n c t i o n w i t h t h e d e -

p e n d e n c e o f k T o n eL~C,, l e a d s t o a s m a l l e r c e l l l i f e t i m e . A s

k,,,.G

i s r e d u c e d , C c, - .~ a p p r o a c h e s c c h a n d t h e i n f l u e n c e

o f l o c a l m a s s t r a n s p o r t d i m i n i s h e s . W i t h k ,i, .a < 1 07 c m 3 /

m o l 9 a n d k,~.~ = 1 06 c m V m o l - s , m o d e l p r e d i c t i o n s a r e

i n d e p e n d e n t o f d i f f u s i o n w i t h i n e a c h p o r e o f t h e e le c -

t r o d e . F o r k H,.N = 1 09 c m 3 / m o l 9 s , l o c a l m a s s t r a n s p o r t s t i l l

i n f l u e n c e s t ee u w h e n kIH.Q i s s m a l l b e c a u s e Cc, a n d Cc,- .w

d o n o t b e c o m e e q u a l ( s e e F i g . 5) .

O v e r a l l, t h e r e s u l t s p r e s e n t e d i n F ig . 6 i n d i c a t e t h a t , f o r

t h e L i /S O C 1 2 s y s t e m , a c h a n g e i n p r e c i p i t a t i o n r a t e c o n -

s t a n t c a n l e a d t o a s i g n i f i c a n t c h a n g e i n c e l l l i f e t im e .

S i m i l a r r e s u l t s a r e e x p e c t e d f o r o t h e r c e l l s w h e r e t h e

m a x i m u m d i s c h a r g e t i rd e i s d i c t a t e d b y t h e p l u g g i n g o f

p o r e s w i t h s o l i d r e a c t i o n p r o d u c t s . F u r t h e r m o r e , i t i s

c o n c e i v a b l e t h a t m a t e r i a l s a d d e d t o e i t h e r t h e e l e c t r o d e

m a t r i x o r t h e e l e c t r o l y t e c o u l d a l t e r t h e r a t e o f n u c l e a -

t i o n a n d g r o w t h o f t h e s o l i d s a n d , h e n c e , h a v e a s u b s t a n -

t i a l e f f e c t o n c e l l p e r f o r m a n c e .

F i g u r e 8 s h o w s t h e o r e t i c a l d i s c h a r g e c u r v e s o b t a i n e d

u s i n g a f i n i t e v a l u e f o r t h e l o c a l m a s s t r a n s p o r t c o e ff i -

c i e n t. C u r v e d is i d e n t i c a l t o t h e c o r r e s p o n d i n g r e s u l t i n

F i g . 2 b e c a u s e C c~-.w = C c ,- a t a l l l o c a t i o n s a n d t h r o u g h o u t

t h e d i s c h a r g e . T h e g e n e r a l s h a p e s o f c u r v e c i n F i g . 2 a n d

8 a r e t h e s a m e . I n p a r t i c u l a r , t h e l o c a l v o l t a g e m i n i m u m

a t t h e o n s e t o f r e a c t i o n [ II ] i s s t i l l p r e d i c t e d , b e c a u s e

cc~-.w g o e s t h r o u g h a m a x i m u m a n a l o g o u s t o t h e c h a n g e s

i n b u l k c o n c e n t r a t i o n s h o w n i n F i g . 1. I n c a s e b , h o w -

e v e r , c h a n g e s i n cc~-.w a r e a t t e n u a t e d ( c o m p a r e F i g . 1 a n d

7 ) a n d a v o l t a g e m i n i m u m i s n o t o b t a i n e d . T h i s i n d i c a t e s

t h a t o n l y c e r t a i n c o m b i n a t i o n s o f t h e n u c l e a t i o n a n d

g r o w t h r a t e c o n s t a n t s l e a d to a v o l t a g e m i n i m u m ; ktH.c

> > kHLN i s a n e c e s s a r y b u t n o t s u f f i c i e n t c o n d i t i o n f o r a

l o c a l m i n i m u m t o o c c u r .

F o r c a s e a i n F i g . 8, t h e c h l o r i d e i o n c o n c e n t r a t i o n i s

s u b s t a n t i a l l y h i g h e r a t t h e w a ll t h a n t h o s e w i t h kc~- = o0.

C o n s e q u e n t l y , t h e v o l t a g e is a p p r o x i m a t e l y 1 80 m V

&

o ~ 2 . 5

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0

k ~ , G

c m a / r n o l " s e c )

a ; 1 0 1 5

b : 1012

c : 10 g

d : 1 0 5

k ] ~' N = 1 0 6 c m a / m ~ 1 7 6 d

a , b

i r i i

1 2 3 4 5

t ( r a i n )

F i g . 8 . T h e o r e t i c a l d i s c h a r g e c u r v e s fo r f o u r v a lu e s o f k m ,G . P a r a m e -

t e r s a r e a s in F i g . 5 . S y m b o l V i n d i c a t e s t h e i n i t i a t i o n o f r e a c t i o n [ 1 1 ].

s m a l l e r t h a n t h e r e s u l t f o r c a s e a i n F ig . 2 . T h e c o r r e -

s p o n d i n g v o l t a g e r e d u c t i o n d u e t o l o c a l m a s s t r a n s p o r t

f o r c a s e s b , c , a n d d a r e 1 2 5 , 4 0 , a n d 0 m V , r e s p e c t i v e l y .

C e l l f a i l u r e i s c a u s e d b y r a p i d r e d u c t i o n o f t h e e f f e c-

t i v e e l e c t r o l y t e c o n d u c t i v i t y a s a r e s u l t o f L iC 1 p r e c ip i t a -

t i o n n e a r t h e f r o n t f a c e o f t h e c a t h o d e (7 ). T h e n o n u n i -

f o r m n a t u r e o f t h e L i C 1 d i s t r i b u t i o n i s i l l u s t r a t e d i n F i g .

9 . T h e c u r v e s c o r r e s p o n d t o t h e c u m u l a t i v e r e a c t i o n r a t e

d i s t r i b u t i o n s f o r r e a c t i o n [ I I] b e c a u s e r e d i s t r i b u t i o n o f

C 1- i o n s i s n e g l i g i b l e . I n g e n e r a l , t h e r e s u l t s i n d i c a t e t h a t

t h e l e ss u n i f o r m t h e r e a c t i o n d i s t r i b u t i o n , t h e s h o r t e r t h e

c e l l l i f e t i m e . A l s o , f a i l u r e o c c u r s a t t h e f r o n t o f t h e e l e c -

t r o d e i n a ll ca s e s . I n c l u s i o n o f l o c a l m a s s t r a n s p o r t o f C 1-

i o n s t e n d s t o m a k e t h e r e a c t i o n d i s t r i b u t i o n m o r e u n i -

f o r m a n d m o r e L i C1 i s d e p o s i t e d t o w a r d t h e b a c k o f t h e

e l e c t r o d e t h a n i f L iC 1 i s a s s u m e d t o b e a d i r e c t p r o d u c t o f

t h e e l e c t r o c h e m i c a l r e a c t i o n (e .g . , c o m p a r e t h e r e s u l t f o r

t h e o l d m o d e l w i t h s o l i d c u r v e s a , b , a n d d ).

T h e r e s u l t s p r e s e n t e d i n F ig . 7 -9 u s e a s p e c i fi c v a l u e f o r

t h e l o c a l m a s s t r a n s p o r t c o e f f i c i e n t k c , -. T h i s v a l u e w a s

e s t i m a t e d u s i n g a n a v e ra g e p o r e d i a m e t e r ( d u r i n g d i s-

c h a r g e ) o f 0. 5 x 1 0 -5 c m ( 2 6) , a n d a b u l k d i f f u s i o n

c o e f f i c i e n t o f 1 0 -5 c m 2 /s . H o w e v e r , t h e e s t i m a t e i s a p -

p r o x i m a t e , a n d i t i s n e c e s s a r y t o a s s e s s t h e s e n s i t i v i t y o f

t h e r e s u l t s t o t h e m a g n i t u d e o f k c, -. F i g u r e 1 0 s h o w s t h a t

s i m i l a r t r e n d s i n b e h a v i o r a r e p r e d i c t e d f o r a w i d e r a n g e

o f k cl - v a l u e s . A s k c , - is i n c r e a s e d , l o c a l m a s s t r a n s p o r t

0 . 8

k c i - = 2 0 c r n / s e c

k]]Z,N = 1 0 6

cm3/mol~

0 . 6 I ~ k m ' G ( C m 3 / r r l ~ e c )

t t~ a , b a ; 1015

t l' , ; / 0 1 o 1

o : 1 o ~

0 . 4 & \ ~ / / c a : l O 5

0 , 2 ,

0 . 0 O . 1

0 . 2 0 . 3

D i s t a n c e T o T h e F r o n t O f P o s it i ve E l e c t r o d e ( m m )

F i g . 9 . D i s t r i b u t i o n o f e l i o i n t h e p o s i t i v e e l e c t r o d e a t t w o d i f f e r e n t

t i m e s f o r f o u r v a l u e s o f k m , c . P a r a m e t e r s a r e a s in F i g . 5 . - - - - - - : t =

1 0 0s . ~ - : e n d o f c e ll d is c ha rg e

i . e . , t c e u =

3 3 4 s f o r a a n d b , te a l1 =

1 7 2 S f o r C , a n d t c el ~ = 2 6 4 s f o r d ) . - - - - : r e s u l t o b t a i n e d u s i n g p r e v i o u s

m o d e l ( 8 ) a t th e e n d o f c e l l d i s c h a rg e ( t ~ l = 2 4 3 s ) .




8/13

V o l . 1 3 3 , N o . 1 1

2

v

u

o

4

P R E C I P I T A T I O N I N E L E C T R O C H E M I C A L C E L L S

2

10 5 10 8 1011

k i n , G ( c r n 3 / m o l 9s e c )

F ig . 1 0 . D e p e n d e n c e o f c e l l li f e ti m e o n k m , c f o r d i f f e r e n t l o c a l m a s s -

t r a n s f e r c o e f f i c i e n t s . P a r a m e t e r s a r e a s i n F i g . 5 .

b e c o m e s l e s s i m p o r t a n t , a n d t h e a s y m p t o t e f o r kc~- = oo i s

a p p r o a c h e d . I n g e n e r a l , t h e s m a l l e r k cl -, t h e l e s s u n i f o r m

t h e d i s t r i b u t i o n s o f cc~-,w a n d U . .w a n d , h e n c e , t h e m o r e

u n i f o r m t h e r e a c t i o n d i s t r i b u t i o n . T h i s r e s u l t s i n a n i n -

c r e a s e i n c e l l l i f e t im e , b u t i t o c c u r s a t t h e e x p e n s e o f c e l l

v o l t a g e ( s e e d e s c r i p t i o n o f F i g . 8).

F o r c o n d i t i o n s w h e r e l o c a l m a s s t r a n s p o r t i s s ig n i f i -

c a n t , i t i s i n t e r e s t i n g t o c o n t r a s t t h e e f f e c t s o f r e d u c i n g

e i t h e r k c , - o r k ,,,.G o n tc~H . I n b o t h c a s e s , t h e w a l l c o n c e n -

t r a t i o n i s i n c r e a s e d , a n d t h i s r a i s e s i o,, a n d l o w e r s U ,,.~ . A

r e d u c t i o n i n

k c , -

a l s o m a k e s t h e d i s t r i b u t i o n o f U ,,,w l e s s

u n i f o r m , a n d t h i s e f f e c t o v e r c o m e s t h e i n c r e a s e i n iou a n d

i s r e s p o n s i b l e f o r t h e i n c r e a s e i n tCeH. O n t h e o t h e r h a n d , a

r e d u c t i o n i n k,,~.G d o e s n o t a l t e r t h e U ..w d i s t r i b u t i o n n o -

t i c e a b l y , a n d t h e i n c r e a s e i n ion d o m i n a t e s , l e a d i n g t o a

s h o r t e r c e l l l i f e t i m e . F i g u r e 1 0 i n d i c a t e s t h a t t h e l o c a l

m a s s t r a n s p o r t o f a s p a r i n g l y s o l u b l e e l e c t r o c h e m i c a l r e-

a c t i o n p r o d u c t c a n h a v e a s ig n i f i c an t e f f e c t o n th e l o c a l

r e a c t i o n r a t e a n d , h e n c e , o n t h e d i s t r i b u t i o n o f t h e p r e -

c i p i ta t e . I n p a r t i c u l a r , s l o w e r d i f f u s i o n w i t h i n a p o r e c a n

i n c r e a s e c e l l l i f e t i m e a p p r e c i a b l y u n d e r s o m e c i r c u m -

s t a n c e s . I n t h e L i /S O C I = c e l l, t h i s c h a n g e w o u l d b e

a c c o m p a n i e d b y a r e d u c t i o n i n t h e v o l ta g e p l a t e a u f o r

t h e n e u t r a l r e a c t i o n , b u t t h e c e l l a l w a y s f a il s b y p l u g g i n g

o f t h e e l e c t r o d e f a c e w i t h L i C1 c r y st a l s .

T h e r e a r e m a n y e l e c t r o d e s w i t h s o l u b l e re a c t i o n p r o d -

u c t s t h a t h a v e s o l i d re a c t a n t s ,

e.g. ,

C d , Z n , P b , a n d P b O 2 .

F o r t h e s e s y s t e m s , t h e e l e c t r o c h e m i c a l r e a c t i o n z o n e

g e n e r a l l y p e n e t r a t e s p r o g r e s s i v e l y f a r t h e r i n t o t h e e le c -

t r o d e a s t h e re a c t a n t s a r e c o n s u m e d . C o n s e q u e n t l y , t h e

e l e c t r o d e v o l u m e a v a i l a b l e f o r r e a c t i o n d i m i n i s h e s d u r -

i n g c e l l o p e r a t i o n , a n d t h e l o c a l t r a n s f e r c u r r e n t r e q u i r e d

t o m e e t t h e s p e c i f i e d c u r r e n t d e n s i t y b e c o m e s l a rg e r .

T h i s t e n d s t o r a i s e t h e w a l l c o n c e n t r a t i o n o f t h e s o l u b l e

r e a c t i o n p r o d u c t a n d t o i n c r e a s e t h e l o c a l e l e c t r o d e o v e r -

p o t e n t i a l ( s e e E q . [3 1] ). U n d e r t h e s e c i r c u m s t a n c e s , i t i s

c o n c e i v a b l e t h a t t h e v o l t a g e r e d u c t i o n w i l l b e s u f f i c ie n t

t o c a u s e c e l l f a i l u r e d i r e c tl y , p a r t i c u l a r l y i f t h e l o c a l m a s s

t r a n s p o r t r a t e is lo w . T h i s e f f e c t w o u l d b e d i r e c t l y a n a l o -

g o u s t o t h e b e h a v i o r o f c e l ls i n v o l v i n g s p a r i n g l y s o l u b l e

r e a c ta n t s , w h e r e a lo w c o n c e n t r a t i o n o f e l e c t r o c h e m i c a l

r e a c t a n t a t t h e e l e c t r o d e w a l l c a n l e a d t o c e l l f a i l u r e (9 ).

2 3 0 3

T h e r e s u l ts p r e s e n t e d a b o v e f o c u s e d a t t e n t i o n o n t h e

e f f e c t s o f k in ,N , k i n ., , a n d k c l - o n c e l l b e h a v i o r . H o w e v e r , a

n u m b e r o f o t h e r f a c t o r s c a n h a v e a s i g n i f i c a n t i m p a c t o n

t h e d i s c h a r g e c h a r a c t e r i s t i c s . T h e s e n s i t i v i t y o f m o d e l

p r e d i c t i o n s t o t h e s e f a c t o r s is d i s c u s s e d b e l o w .

M o r p h o lo g y p a r a m e t e r m f o r p r e c i p i t a t i o n .- - T h e

q u a n t i t y m a p p e a r s i n E q . [8 ], a n d i t c o n t r o l s t h e c h a n g e

i n t h e e f f e c t i v e p r e c i p i t a t i o n r a t e c o n s t a n t kT d u e t o a n

i n c r e a s e i n th e s u r f a c e a r e a o f c ry s t a l s a t w h i c h f u r t h e r

g r o w t h c a n t a k e p l a ce . F r o m a p h y s i c a l s t a n d p o i n t , t h e

m a g n i t u d e o f t h e e x p o n e n t m r e fl e c t s t h e m o r p h o l o g y o f

t h e L i C 1 p r e c i p i t a t e : l a r g e v a l u e s c o r r e s p o n d t o c r y s t al s

w i t h s m a l l s u r f a c e / v o l u m e r a t i o s a n d v i c e v e rs a . I n F ig .

1 1, c e l l l i f e t i m e s f o r m = 0 a n d 0 .5 a re c o m p a r e d w i t h t h e

r e s u l t f r o m F i g . 6 , w h i c h u s e d m = 1 .0. T h e a s y m p t o t e f o r

f a s t p r e c i p i t a t i o n o n th e r i g h t - h a n d s i d e o f t h e d i a g r a m i s

i n d e p e n d e n t o f m . A l s o , o n t h e l e f t - h a n d s i d e , t h e kl~l.G

t e r m i n E q . [8 ], a n d h e n c e m , i s n o t i m p o r t a n t , b e c a u s e

k m.~ d o m i n a t e s . T h e s e n s i t i v i t y o f tcen t o m i n t h e i n t e r m e -

d i a t e r e g i o n c a n b e r e l a t e d t o c h a n g e s i n kT. T h e s m a l l e r

t h e v a l u e o f m , t h e g r e a t e r t h e i n c r e a s e i n kT l o c a l l y f o r a

g i v e n a m o u n t o f L iC 1 p r e c i p i t a t io n . T h i s c h a n g e i s e q u i v -

a l e n t t o h a v i n g a l a r g e r km .G v a l u e , b u t w i t h m = 1 .0 . T h e

e f f e c t i s t o l o w e r t h e c h l o r i d e i o n c o n c e n t r a t i o n a n d ,

t h r o u g h c h a n g e s i n io~La n d U H ,w ( s e e E q . [ 3 1] a n d [ 3 2] ), t o

r a i s e t cell. W i t h m = 0 , k v i s c o n s t a n t t h r o u g h o u t t h e d i s -

c h a r g e ( k v = k m .N + k nL G) a n d tceH v a r i e s m o n o t o n i c a l l y b e -

t w e e n t h e a s y m p t o t e s . T h i s s u g g e s ts t h a t t h e m i n i m a f o r

tCe~ w i t h m = 0 . 5 a n d 1 .0 a r e c a u s e d , a t l e a s t i n p a r t , b y t h e

l o c a l c h a n g e s i n k v t h a t t a k e p l a c e a s p r e c i p i t a t i o n p r o -

c e e d s . W i t h m = 1 .0 , a l o c a l v o l t a g e m i n i m u m i s p r e -

d i c t e d f o r 1 06 > - km.JkH ~,~ >- 1 02 . A r e d u c t i o n i n t h e v a l u e

o f t h e e x p o n e n t m r e d u c e s t h e r a n g e o f km,c /km.N f o r a

m i n i m u m . F o r m = 0 , t h e C 1 i o n c o n c e n t r a t i o n a l w a y s

c h a n g e s m o n o t o n i c a l l y d u r i n g d i s c h a r g e , a n d a l o c a l

v o l t a g e m i n i m u m i s n e v e r o b t a in e d .

K i n e t i c p a r a m e t e r a~ f o r r e a c t i o n [ I I ] .- - F i g u r e 12

i l l u s t ra t e s t h e s e n s i t i v i t y o f t h e m o d e l c a l c u l a t i o n s t o t h e

"2

E

v

o

2 [ _ . _

1 0 5

I I I

r n=0

m = 0 . 5

k i l l, N = 1 0 6 c m 3 / m o l . s e c

k T = k i n , N + ( i c i ) m k r r r , G

10 8 1011 1014

k r n , G

( c m 3 / m o l 9s e c )

F i g. 1 1 . D e p e n d e n c e o f c e l l li f e t im e o n k i l z ,G o r t h r e e v a l u e s o f m .

P a r a m e t e r s a r e a s i n F i g . 5 .




9/13

2304 J.

Electrochem Soc :

ELECTROCHEMICAL SCIENCE AND TECHNOLOGY

November 1986

4

E

|

3

I

kTrl ., N = 1 06cm 3 / r no l . s ec

a~ 2 .5 A / c m 3

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i i I

1 0 5 1 0 7 1 0 9

v

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6

I I I

p = 0 . 1

p =0 .5

p = 1 . 0

kr r r , N

= 106cm3/mo I.

s e c

k r n , G ( c m 3 / m o l ' s e c )

F i g. 1 2 . D e p e n d e n c e o f c e ll l i f e t im e o n k n ~ ,c f o r d i f fe r e n t v a l u e s o f

a ~ P a r a m e t e r s a r e a s i n F i g . 5 .

p r o d u c t o f t h e i n i t i a l s p e c i f i c s u r f a c e a r e a a ~ a n d t h e r e f -

e r e n c e e x c h a n g e c u r r e n t d e n s i t y io,,.ref f o r r e a c t i o n [ II ]. I n

g e n e r a l , a r e d u c t i o n i n t h e v a l u e o f t h e p r o d u c t l e a d s t o

g r e a t e r k i n e t i c c o n t r o l a n d t h i s h a s t w o m a i n e f f e c t s o n

c e l l b e h a v i o r : ( i) t h e r e a c t i o n d i s t r i b u t i o n i s m o r e u n i -

f o r m a n d , c o n s e q u e n t l y , t h e c e l l l i f e t i m e is l o n g e r , a n d

( i t )

c h a n g e s i n U ,,.w a r e le s s i m p o r t a n t a n d , h e n c e , t h e r e i s

l e s s t e n d e n c y f o r a m i n i m u m i n t o e~ a s k ~,,.~ i s a l t e r e d .

W i t h s m a l l v a l u e s o f kin .G , t h e r a t e o f L iC 1 p r e c i p i t a t i o n i s

m o r e i m p o r t a n t t h a n t h e r a t e c o n s t a n t f o r r e a c t i o n [ I I ]

a n d , a s a r e s u l t , t h e c u r v e s a r e l e s s s e n s i t i v e t o a~ i n

t h i s r e g i o n . F o r k,H.a = 1 0 ~ c m 3 / m o l 9 s , k i l l , N d o m i n a t e s i n

E q . [8 ] a n d a n u c l e a t i o n - c o n t r o l l e d a s y m p t o t e f o r t h e c e l l

l i f e t i m e i s a t t a i n e d .

C o v e r a g e f a c t o r p 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 s . - - T h e

e x p o n e n t p i n E q . [6 ] i n f l u e n c e s t h e e f f e c t i v e s u r f a c e a r e a

a v a i l a b l e f o r t h e e l e c t r o c h e m i c a l r e a c t i o n s . W i t h s m a l l e r

v a l u e s o f p , t h e c a r b o n s u r f a c e i n t h e c a t h o d e i s m o r e

r e a d i l y p a s s i v a t e d b y L i C 1 c r y s t a l s . T h i s f o r c e s t h e r e a c -

t i o n d e e p e r i n t o t h e e l e c t r o d e a n d d e l a y s f a i l u r e b y p l u g -

g i n g a t t h e f r o n t f a c e o f t h e e l e c t r o d e . F i g u r e 1 3 s h o w s

t h a t l o n g e r l i f e t i m e s a r e p r e d i c t e d f o r s m a l l e r p v a l u e s

b u t t h a t t h e i n f l u e n c e o f k,,,.c is s i m i l a r i n e a c h c a s e . W i t h

s m a l l k ,, , .c v a l u e s , t h e c e l l l i f e t i m e i s l e s s s e n s i t i v e t o p

b e c a u s e t h e e l e c t r o c h e m i c a l r e a c t i o n r a t e is l e s s i m p o r -

t a n t t h a n t h e p r e c i p i t a t i o n r a t e (s e e d e s c r i p t i o n o f F i g .

12) .

W i t h h i g h p r e c i p i t a t io n r a t e s ( i .e. , l a r g e k , m G ) , o n e

m i g h t e x p e c t t h e d e p e n d e n c e o f tr o n p t o m a t c h t h e r e-

s u l t s o b t a i n e d p r e v i o u s l y ( 7) w h e n s o l i d L i C I w a s r e -

g a r d e d a s a n e l e c t ro c h e m i c a l r e a c t i o n p r o d u c t . H o w e v e r ,

t h e r e s u l t s p r e s e n t e d i n F ig . 1 3 a r e r e l a t i v e l y i n s e n s i t i v e

t o t h e v a l u e o f p . F o r e x a m p l e , a n i n c r e a s e i n p f r o m 0 .1

t o 0 . 5 r e d u c e s tr16 2 y 9 % i n F i g . 1 3 b u t b y 3 8 % i n t h e p r e v i -

o u s m o d e l (7 ). T h e d i f f e r e n c e c a n b e a t t r i b u t e d t o t h e in -

c l u s i o n o f l o c a l m a s s t r a n s p o r t , a n d t o t h e d e p e n d e n c e o f

io ,l o n t h e w a l l c o n c e n t r a t i o n C c,-.w r a t h e r t h a n t h e b u l k

c o n c e n t r a t i o n C c,-, i n t h e p r e s e n t a n a l y s i s . W i t h l a r g e

k,,,.c, t h e r a t i o Cc, .w/Cc,- i s m u c h g r e a t e r t h a n u n i t y ( s e e

F i g . 5 ) a n d , h e n c e , t h e e f f e c t i v e e x c h a n g e c u r r e n t d e n s i t y

i s m u c h l a r g e r w i t h t h e p r e s e n t m o d e l . T h i s d i m i n i s h e s

t h e i m p o r t a n c e o f b o t h e l e c t ro c h e m i c a l k i n e t i c s a n d

c h a n g e s i n t h e p r o d u c t a~ a n d , i n t u r n , a t t e n u a t e s

t h e d e p e n d e n c e o f t c e U o n p . I t i s e m p h a s i z e d , h o w e v e r ,

~

1 0 5 1 0 8 1 o ~ ~ 1 o T M

k i ii , G ( c m 3 / r n o l . s e c )

F i g. 1 3 . D e p e n d e n c e o f c e ll l i fe t i m e o n k m , c f o r t h r e e v a l u e s o f p . P a -

r a m e t e r s a r e a s i n F i g . 5 .

t h a t a s k c l - a p p r o a c h e s i n f i n it y , l o c a l m a s s t r a n s p o r t b e -

c o m e s u n i m p o r t a n t , a n d t h e i n f lu e n c e o f p p r e d i c t e d b y

t h e t w o m o d e l s b e c o m e s i d e n ti c a l .

S o l u b i l i t y p r o d u c t , K ~ p . - -W i t h s l o w p r e c i p i t a t i o n , t h e

s u p e r s a t u r a t i o n l e v e l i s h i g h , a n d t h e p r o d u c t ( cL i+ )( cc ,- )

t h a t a p p e a r s i n E q . [7 ] i s m u c h l a r g e r t h a n K sp . F o r l a r g e

k ,, ,.~ , t h e r e s u l t s a r e a l s o i n s e n s i t i v e t o t h e s o l u b i l i t y

p r o d u c t . W i t h k i l l , N = 1 0 6 c m 3 / m o l 9 a n d k ,, ,. c = 1 O s c m 3 /

t o o l 9 s , a h u n d r e d f o l d r e d u c t i o n i n Ksp h a s n o e f f e c t o n

te e,, a n d a h u n d r e d f o l d i n c r e a s e o n l y l o w e r s t ee ,, b y 3 . 6 % .

T h e c h a n g e i s c a u s e d b y a n i n c r e a s e i n C c~- w h i c h r a i s e s

C c, .w s l i g h t l y ( t o s a t i s f y E q . [ 5 ] ) a n d , i n t u r n , r a i s e s to ,,.

T h e r e s u l t s p r e s e n t e d a b o v e , f o r a s o l u b l e r e a c t i o n

p r o d u c t , c a n b e c o n t r a s t e d w i t h t h o s e f o r a s p a r i n g l y s o l-

u b l e r e a c t a n t . I n t h e l a t t e r c a se , t h e b u l k c o n c e n t r a t i o n i s

m u c h l a r g e r t h a n t h e w a ll c o n c e n t r a t i o n a n d t h e m a g n i -

t u d e o f t h e b u l k c o n c e n t r a t i o n c a n d e t e r m i n e t h e m a s s

t r a n s p o r t r a t e t o t h e r e a c t i o n s i t e a n d h e n c 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 ( 5, 9 ). T h e r e f o r e , t h e s o l u b i l i t y o f t h e

r e a c t a n t c o u l d s i g n i f i c a n t l y a ff e c t c el l b e h a v i o r .

C e ll t e m p e r a t u r e . - - T h e t e m p e r a t u r e o f a Li /S O C 1 2 c e l l

u s u a l l y r i s e s d u r i n g d i s c h a r g e . A t h i g h e r t e m p e r a t u r e s ,

t h e e l e c t r o l y t e c o n d u c t i v i t y K~ a n d t h e s p e c i e s d i f f u s i o n

c o e f f i c i e n t s , D ~k, a r e l a r g e r , a n d t h e s e f a c t o r s , t o g e t h e r

w i t h t h e te m p e r a t u r e - d e p e n d e n t t e r m s i n E q . [1 ], t e n d t o

m a k e t h e r e a c t i o n d i s t r i b u t i o n m o r e u n i f o r m . I n g e n e r a l ,

i t i s a l s o n e c e s s a r y t o a c c o u n t f o r t h e t e m p e r a t u r e d e p e n -

d e n c e o f k i n e t i c p a r a m e t e r s , s u c h as io I 1 , r e t , kin,N, kul,a, m ,

a n d p . H o w e v e r , i n f o r m a t i o n o n a c t i v a t i o n e n e r g i e s f o r

t h e s e p a r a m e t e r s i s n o t a v a i l a b l e . F o r t h e c e ll s c o n s i d -

e r e d h e r e , t h e h e a t t r a n s f e r c o e f f i c i e n t is l a r g e e n o u g h t o

r e s t r i c t t e m p e r a t u r e r i s e s t o 1 0~ 17 6 a n d , t h e r e f o r e , t h e

e f f e c ts o f t e m p e r a t u r e c h a n g e s a r e r e l a t i v e ly u n i m -

p o r t a n t .

C o r r o s i o n p o t e n t i a l . - - I n t h e L i / SO C 1 2 s y s t e m , i t m a y

b e m o r e a p p r o p r i a t e t o r e g a r d t h e o p e n - c i r c u i t p o t e n t i a l s

a n d e x c h a n g e c u r r e

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