Download pdf - A Mathematical Model of Crevice and Pitting Corrosion the Physical Model

7/21/2019 A Mathematical Model of Crevice and Pitting Corrosion the Physical Model

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Co rrosion Scien ce, Vo l. 28, No. 6, pp. 603-6 20, 1988 0010--938X/88 $3.(~1+ 0.00

Printed in Gre at Britain Pergamo n Press plc

A M A T H E M A T I C A L M O D E L O F C R E V I C E A N D P I T T I N G

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

S. M . SHARLANDan d P. W . TASKER

T h e o r e t i c a l P h y s i c s D i v i s i o n , H a r w e l l L a b o r a t o r y , O x f o r d s h i r e O X 1 1 0 R A , U . K .

A b s t r a c t - - A p r e d i ct i v e a n d s e l f- c o n si s te n t m a t h e m a t i c a l m o d e l i n c o r p o ra t i n g t h e e l e c t r o c he m i c a l ,

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

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

k i n e t i c s ( a n d h e n c e m e t a l p e n e t r a t i o n r a t e s ) w i t h i n a n a c ti v e c o r ro s i o n c a v i t y a s a f u n c t i o n o f t h e m a n y

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

c r ev i ce i s m o d e l l ed a s a p a r a l l e l - s i d ed s l o t f i l l ed w i t h a d i l u t e s o d i u m ch l o r i d e s o l u t i o n . T h e cav i t y

p r o p a g a t i o n r a t e s a r e f o u n d t o b e f a s t e r i n t h e c a s e o f a c r e v i c e w i t h p a s s iv e w a l ls t h a n o n e w i t h a c t iv e w a ll s.

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

t h i s m o d e l , g i v in g a n i n d i c a t i o n o f t h e f u t u r e s h a p e o f t h e c a v i ty . T h e m o d e l i s e x t e n d e d t o i n c l u d e a s ol i d

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

p h y s i c a l e n v i r o n m e n t w i t h i n t h e c r e v i c e o n t h e p r e d i c t e d c o r r o s i o n r a t e s . I n t h is p a p e r , t h e m o d e l i s

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

N O M E N C L A T U R E

i

c o n c e n t r a t i o n o f c h e m i c a l s p e c ie s , i, m o l d m 3

D i d i f f u s i o n co e f f i c i en t o f s p ec i e s i , m e s - 1

F F a r a d a y ' s c o n s t a n t o f e l e c t ro l y s is , C m o l -~

i cu r r e n t d en s i t y , A m - 2

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

K e q u i l i b r i u m c o n s t a n t

l en g t h o f c r ev i ce , m

R g m o l a r g a s c o n s t a n t , J m o V 1 K

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

T t e m p e r a t u r e , K

U i m o b i l i t y o f s p ec i e s , m 2 m o l J - ~ - 1

w w i d t h o f c r e v i c e , m

z , c h a r g e n u m b e r o f s p e c i e s i

c t e l ec t r o ch em i ca l p a r t i t i o n co e f f i c i en t

q~ e l ec t r o s t a t i c p o t e n t i a l d r o p i n c r ev i ce , V

~0M e l e c , r o s t a t ic p o t e n t i a l o f c o r r o d i n g m e t a l , V

a ch a r g e d en s i t y , C m -3

I N T R O D U C T I O N

A P A S S I V E m e t a l s u r f a c e e x p o s e d t o a c o r r o s i v e e n v i r o n m e n t m a y e x p e r i e n c e a t t a c k

a t s e v e r a l i s o l a t e d s i te s . I f t h e t o t a l a r e a o f t h e s e s i t e s i s s m a l l c o m p a r e d w i t h t h e

w h o l e s u r f a c e a r e a , t h e n t h e m e t a l i s s a i d t o b e e x p e r i e n c i n g l o c a l i z e d c o r r o s i o n . T h e

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

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

S e v e r a l d i f f e r e n t m o d e s o f l o c a l i z e d c o r r o s i o n m a y b e i d e n t i f i e d a n d a r e d e p e n d e n t

M an u s c r i p t r ece i v e d 1 5 J an u a r y 1 9 86 ; i n r ev i s ed f o r m 1 5 A u g u s t 1 9 87 .

603


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604 S .M . SHARLAND nd P. W . TASKER

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

a t t h e t i m e o f a t t a c k . O n e o f t h e m o s t f a m i l i a r is p i t ti n g c o r r o s i o n w h i c h is

c h a r a c t e r i z e d b y t h e p r e s e n c e o f a n u m b e r o f sm a l l, r o u g h l y h e m i s p h e r i c a l c a vi ti e s

o n t h e e x p o s e d s u r f a c e . T h e g e o m e t r i e s o f t h e s e p it s d e p e n d o n m a n y f a c t o r s s u c h as

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

c r e v ic e a t t a c k w h i c h o c c u r s w h e r e t h e r e a r e t w o o r m o r e s u r fa c e s i n c l os e p r o x i m i t y ,

l e a d in g t o t h e c r e a t i o n o f a l oc a l ly o c c l u d e d r e g i o n w h e r e e n h a n c e d d i s so l u ti o n m a y

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

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

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

b e c o n s i d e r e d h e r e .

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

s t r u c t u r e h a s l e d to a g r e a t d e al o f s t u d y o f th is p h e n o m e n o n . T h e r e a r e m a n y

d i ff ic u lt ie s a s s o c i a te d w i t h t h e e x p e r i m e n t a l m e a s u r e m e n t o f t h e e l e c t r o c h e m i c a l

c o n d i t io n s w i t h in c a v i ti e s a n d a s a r e su l t a n u m b e r o f m a t h e m a t i c a l m o d e l s h a v e b e e n

d e v e l o p e d . T h e a i m o f th is p a p e r i s t o p r e s e n t a m o d e l o f th e p r o p a g a t i o n o f a n a c t iv e

c o r r o s i o n c r e v i c e o r p i t w h i c h i s e n t i r e l y p r e d i c t i v e a n d s e l f - c o n s i s te n t , u s i n g a s e r ie s

o f ph y s ic a l ly b a s e d a p p r o x i m a t i o n s a n d a s s u m p t io n s . T h e m o d e l p r e d i c ts t h e s t e a dy -

s t a te s o l u ti o n c h e m i s t r y a n d e l e c t r o c h e m i s t r y ( a n d h e n c e m e t a l p e n e t r a t i o n r a t e s )

w i t h in t h e r e s t r i c t e d g e o m e t r i e s o f th e c a v it ie s . T h e i n p u t p a r a m e t e r s o f t h e m o d e l

d e s c r i b e t h e p h y si c a l a n d c h e m i c a l e n v i r o n m e n t o f t h e c r e v i c e , e . g . m e t a l p o t e n t i a l ,

p H o f th e e l e c t r o l y t e e t c. T h e m o d e l a ls o us e s e m p i r i c a ll y d e t e r m i n e d e l e c t r o d e

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

a d d i t io n a l p h y s ic a l p r o c e s s o r c h e m i c a l e q u il i b r iu m r e a c t i o n . T h e r e l a t e d m a t h e m a t -

i ca l a d d i t i o n s , j u s t if i c a t io n o f a p p r o x i m a t i o n s a n d m e t h o d s o f s o l u t i o n a t e a c h s t a g e

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

c o m p a r i s o n o f t h e p r e d i c t i o n s o f t h e m o d e l w i th e x p e r i m e n t a l r e s u lt s in t hi s p a p e r ,

b u t m o r e g e n e r a l a s s e s s m e n t s o f th e a c c u r a c y o f t h e m o d e l w i ll be m a d e b y

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

q u a l i t a t i v e v a r i a t i o n in t h e s e w i t h t h e p a r a m e t e r s o f t h e s y s t e m . I n t h is w a y , t h e

i m p o r t a n c e o f s o m e o f t h e a p p r o x i m a t i o n s a n d e m p i r ic a l i n p ut d a t a t o t h e p re d i c t i o n s

o f t h e m o d e l m a y b e d e t e r m i n e d . A m o r e d e t a i l e d m o d e l v a li d a t io n w ill b e c a r r ie d

o u t a t a l a t e r s t a g e . I n t hi s p a p e r , i n p u t d a t a a p p r o p r i a t e t o th e c o r r o s i o n o f c a r b o n

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

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

p i t ti n g o r c r e v i c e c o r r o s i o n ) i n a w i d e r a n g e o f s o l u t io n s c o n t a i n i n g c h l o r i d e i o ns ~ ).

I n t h e f ir st s e c t io n , t h e m a t h e m a t i c a l d e s c r i p t i o n o f a n a c t i v e c r e v i c e o r p it is

d i sc u s se d . T h e s e m e t h o d s w ill b e c o m p a r e d w i th s e v e r a l m o d e l s o f th e s o l u t io n

c h e m i s t r y a n d e l e c t r o c h e m i s t r y w i t h in c o r r o d i n g c a v i ti e s fr o m t h e l i t e r a t u re . I n t h e

f o l l o w i n g s e c t i o n , d e t a il s o f a s im p l e m o d e l o f a c r e v i c e w h i c h i s c o r r o d i n g o n l y a t t h e

b a s e a n d i n c l u d e s o n l y o n e h y d r o l y s i s r e a c t i o n a r e o u t l i n e d . T h e w a l ls o f t h is c a v i ty

a r e a s s u m e d p a s s i v e , a l l o w i n g a s i m p l if i ed d e s c r i p t i o n o f t h e i o n i c m i g r a t i o n p r o c e s -

s e s w i t h i n t h e c a v i t y . T h i s m o d e l is t h e n e x t e n d e d t o a m o r e r e a l is t ic d e s c r i p t i o n o f a n

e s t a b l i s h e d l o c a l i z e d c o r r o s i o n s i te a n d i t is a s s u m e d t h a t c o r r o s i o n o c c u r s o n t h e

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

c o m p a r e d . I n p a r t i c u l a r , t h e d i s t r i b u t io n o f c u r r e n t o n t h e w a ll s is c o n s i d e r e d a n d t h e

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

t h e c a v i t y .


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A m o d e l o f c r e v ic e a n d p i t ti n g c o r r o s i o n - - I 6 0 5

Precipitation of solid hydroxide is monitored in both these models by checking

the concentrations for appropriate excesses above the solubility limit of the solid. In

the next section, the active and passive wall models are extended to include the

precipitation reaction directly. The kinetics of the precipitation are assumed to be

rapid compared with the migration processes and the ionic concentrations are

constrained by the equilibria relation for the precipitation reaction. The results are

compared with those from a crevice with no solid present. The sensitivity of the

predictions to the diffusion coefficient in the crevice (reduced by the presence of the

solid corrosion product) is tested. In the final section, limitations of the models and

possible extensions to improve agreement with experiment are discussed.

M A T H E M A T I C A L D E S C R I P T I O N O F C R E V I C E A N D P I T P R O P A G A T I O N

The chemical environment within a localized corrosion cavity is often very

different from that outside. The common aim of the mathematical models that are

described here is the prediction of the solution chemistry and electrochemistry within

the restricted geometries of the cavities as a function of the many physical and

chemical parameters of the system, such as crevice dimensions and bulk solution

composition. Such information yields metal penet ration rates. A full description of

the problem should include an account of the complex solution chemistry within the

crevice, the electrochemical reaction rates and their dependence on parameters such

as electrostatic potential and solution pH, the migration of ions under both concen-

tration and potential gradients, the advection of ions in flowing electrolytes, the

effect of the changing shape and dimensions of the crevice as propagation proceeds

and the effect of the blocking of the crevice with solid corrosion product. These

processes are schematical ly illustrated in Fig. 1.

Moving boundary

Cathodic

J processes

dffusn

. . . lu t lon J P r e c i p l t o t l o n

I o n

IT

C o n c e n t r a t i o n g r a d i e n t s )

[ C h e m i c a l r e a c t i o n ] E l e c t r o m u g r a t u o n

p o t e n t i a l g r a d i e n t )

Y

l

F IG . 1 . S c h e m a t i c i ll u s t r a t i o n o f t h e c re v i c e o f p i t m o d e l .


4/18

606 S.M. SHARLANDand P. W. TASKER

T h e m a t h e m a t i c a l m o d e l o f c av i ty p r o p a g a t i o n is b a s e d o n t h e s e t o f f u n d a m e n t a l

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

e l e c t r o l y ti c s o lu t i o n s . T h e c o n c e n t r a t i o n o f c h e m i c a l s p e c i e s i, C i o b e y s t h e f o l lo w i n g

m a s s - c o n s e r v a t i o n e q u a t i o n ( a s s u m i n g t h e e l e c t r o l y t e is s t a ti c ),

O C i - D i V 2 C i + z i U i V C i ~ ) ) q- R i . (1)

Ot

D i i s t h e e f f e c t i v e d i f f u s i o n c o e f f i c i e n t o f s p e c i e s

i , zi

t h e c h a r g e n u m b e r , ~ i s t h e

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

d e p l e t i o n o f s p e c i e s i b y c h e m i c a l r e a c t i o n a n d U i i s t h e m o b i l i ty , g i v e n b y t h e

e x p r e s s i o n

D i

U i - R T (2)

T h e f ir st t e r m o n t h e r i g h t- h a n d s i d e o f th e e q u a t i o n r e p r e s e n t s t h e c o n t r i b u t i o n t o

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

b y e l e c t r o m i g r a t i o n f o r t h e c h a r g e d s p e c i e s. T h e e l e c t r o s t a ti c p o t e n t i a l o f t h e s y s t e m ,

is g o v e r n e d b y P o i s s o n ' s e q u a t i o n ,

v2q~ = ~ , (3)

w h e r e a is t h e c h a r g e d e n s i t y a n d e is t h e p e r m i t ti v i t y o f th e e l e c t r o l y t e . F o r w a t e r e

i s 8 0 , a n d t h e m a g n i t u d e o f e -1 is s u f f ic i e n t l y l a r g e t h a t a n y d e p a r t u r e o f t h e s y s t e m

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

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

w i t h t h e d i f f u s i o n p r o c e s s e s . A s a n a p p r o x i m a t i o n t h e r e f o r e , P o i s s o n ' s e q u a t i o n i s

o f t e n r e p l a c e d b y t h e e q u a t i o n o f lo c a l c h a r g e n e u t r a li t y

~ i z i C i x ) ~--- 0 (4)

f o r al l x i n t h e c r e v i c e . T h e b o u n d a r y c o n d i t i o n s o f t h e p r o b l e m a r e a s f o ll o w s :

( 1) T h e c o n c e n t r a t io n s o f th e s p e c i e s a r e f ix e d at t h e c a v i ty m o u t h a n d a r e e q u a l

t o t h e v a l u e s i n t h e b u l k s o l u t i o n o u t s i d e t h e c o r r o s i o n s i te .

( 2) T h e f lu x o f s p e c ie s e i t h e r p r o d u c e d o r c o n s u m e d i n th e e l e c t r o d e p r o c e s s e s a re

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

w a l l s. T h e f lu x e s o f t h e o t h e r s p e c i e s a t t h e w a l l s a r e z e r o .

S o l u t i o n s o f t h e p r o b l e m h a v e b e e n o b t a i n e d u s in g a v a r i e t y o f a p p r o x i m a t i o n s t o

t h e e q u a t i o n s t h e m s e l v e s , t h e b o u n d a r y c o n d i ti o n s a n d t h e g e o m e t r y o f t h e c r ev i ce .

T h e m o s t c o m m o n a p p r o x i m a t i o n s a r e a s f o ll o w s :

( 1) A d o p t i o n o f si m p l if ie d g e o m e t r i e s r e p r e s e n t i n g t h e p i t o r c re v i c e .

(2 ) R e d u c t i o n o f t h e n u m b e r o f d i m e n s i o n s o f t h e p r o b l e m , u s u a ll y t o o n e , f o r

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

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

c a v i t y w a l l s a r e p a s s i v e .

( 3 ) N e g l e c t o f d i f fu s i o n o f i o n ic s p e c i e s u n d e r c o n c e n t r a t i o n g r a d i e n ts .

( 4) N e g l e c t o f m i g r a t io n o f c h a r g e d s p e c i e s u n d e r p o t e n t i a l g r a d i e n ts .

( 5 ) N e g l e c t o f c o n v e c t i o n o f s p e c ie s b y a m o v i n g e l e c t r o l y t e .


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A modelof crevice and pitting corrosion--I 607

(6) Neglect of the effects of the retreating walls of the cavity, i.e. the convected

electrolyte and the lengthening of the migration paths of the species.

(7) Simplification of the chemistry and electrochemistry by consideration of a

small number of chemical species and a limited number of equilibria and electrode

reactions.

(8) Neglect of the dependence of electrochemical reaction rates on solution

chemistry and electrochemistry within the crevice.

(9) Assumption of the steady state.

The simplest models consider ionic migration by one transport mode only, but

these are only strictly applicable to fairly narrow ranges of timescales and environ-

ments. For example, models which neglect ionic diffusion are only valid at short

times since any increase in electrostatic potential in the cavity implies some imbalance

in ionic concentrations and subsequent diffusion. Similarly, models which neglect

electromigration are also restricted; they are either specific to some experimental

system in which the potential gradients have been shown to be small over the relevant

timescales or they consider only the transport of a neutral species. The more general

models are those which consider diffusion and electromigration (and convection for

situations in which the electrolyte is flowing). There have been a number of such

models reported in the literature.

Some of the methods used in two literature models (Turnbull et al. 2-3 and Galvele

et al. 4-6) are now compared and the importance of some of the assumptions and

approximations to the general accuracy of the predictions are discussed. These

models involve a parallel-sided corroding cavity and yield a steady-state solution for

the chemistry and electrochemistry although the applications of the models do differ.

Both models consider diffusion, electromigration and chemical reaction of at least six

aqueous species within the cavity. This is the minimum number necessary to simulate

the behaviour of a localized corrosion site. These species are metal ions from the

dissolution process (MeZ+), hydroxyl and hydrogen ions from the water dissociation

(O H- and H+), a metal hydrolysis product (MeOH ~z- l)+ as the simplest) and sodium

and chloride ions to facilitate current flow within the crevice. Both models include

some buffering reactions to obtain more realistic simulation of particular experimen-

tal systems. However there are a number of differences; Galvele considers a cavity

with passive walls, whereas Turnbull includes the effect of corrosion on the walls.

Also, there are differences in the approximations made to the boundary conditions

of the problem and in the range of parameters over which each model is applied. The

models of Turnbull are applied to conditions of cathodic control [for metals polarized

to potentials less than about -0 .5 V(SCE)]. The anodic dissolution of the metal is

assumed to follow Tafel kinetics, i.e. the dissolution current is assumed to obey the

following equation ,

i = i0 exp - a E / R T ) , (5)

with E the electrode potential and the parameters i0 and a obtained from polarization

experiments. Two cathodic reactions are considered in this model; the reduction of

water (also assumed to obey Tafel kinetics) and the reduction of hydrogen ions. The

rate of the latter reaction is assumed to depend linearly on the pH, i.e.

i = i0[H +] exp - a E / R T ) . (6)


6/18

608 S .M . SHARLAND nd P. W . TASKER

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

F e 2+ + H 2 0 ~ F e O H + + H + (7 )

s i n ce n o h y d r o x i d e is a s s u m e d t o p r e c i p i t a t e a t l o w m e t a l p o t e n t i a l s .

I n t h e m o d e l s o f G a l v e l e

e t a l .

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

n e g l e c t e d . T h i s a p p r o x i m a t i o n l im i ts t h e r a n g e o f a p p l ic a t i o n t o h i g h e r m e t a l

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

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

h o w e v e r , t h e k i n e ti c s o f t h e a n o d i c d i s s o lu t i o n r e a c t io n a r e a s s u m e d , t h i s m o d e l

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

d e n s i ty , w h e r e a s t h e m o d e l o f T u r n b u l l c a n b e u s e d t o p r e d i c t t h e c u r r en t f ro m t h e

m e t a l p o t e n t i a l . T h e m o d e l o f G a l v e l e a l s o in c l u d e s t h e p r e c i p i t a ti o n o f a s o li d

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

e x c e e d t h e s o l u b i l i ty p r o d u c t o f t h is p h a s e . T h e s o li d is t r e a t e d a s a m o b i l e s p e c i e s

a n d t h e a p p r o p r i a t e m a s s - b a l a n c e e q u a t i o n i s d e r i v e d a n d s o l v e d . T h e h y d r o x i d e i s

a s s u m e d t o b e p r e c i p i t a t e d a s c o l l o id a l p a rt i c le s a n d n o t t o i n f l u e n c e t h e m i g r a ti o n o f

o t h e r m o b i l e s p e c i e s ( t h r o u g h s c al in g o f t h e d i f f u s io n c o e f f ic i e n t s, fo r e x a m p l e ) . T h i s

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

t i o n i n a c o r r o d i n g d e v i c e .

T h e p r e d i c ti o n s o f b o t h m o d e l s h a v e b e e n c o m p a r e d w i t h v a r io u s e x p e ri m e n t s .

T u r n b u l l c o m p a r e d r e s u l ts f ro m h i s m o d e l w i th m e a s u r e d p o t e n t ia l s a n d p H v a l u e s

i n an a r ti fi ci al c re v i c e . H e o b t a i n s g o o d q u a l i ta t i v e a g r e e m e n t w i t h e x p e r i m e n t a l d a t a

b u t b o t h t h e p r e d i c t e d h y d r o g e n i o n c o n c e n t ra t i o n a n d t h e p o t e n t i al d r o p i n t h e

c a v it y ar e s o m e w h a t l o w e r t h a n m e a s u r e d e x p e r i m e n t a l r e s ul ts . A l t h o u g h n o

m e a s u r e m e n t s o f th e c a v i ty p r o p a g a t i o n r a te s a r e g i v e n, s o n o d i re c t c o m p a r i s o n m a y

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

G a l v e l e t e s t e d h i s m o d e l b y c o m p a r i n g a c a l c u l a t e d p it t in g p o t e n t i a l a g a i n s t e x p e r -

i m e n t . T h e a g r e e m e n t w a s fo u n d t o b e g o o d f o r a n u m b e r o f d i f fe r e n t s o l u ti o n

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

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

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

f a s t e r t h a n r e a s o n a b l y e x p e c t e d f o r s u c h a p o l a r i z a ti o n . T u r n b u l l s u g g e s t s t h a t t h e

i n a c c u r a c y i n s o m e o f h i s p r e d i c t i o n s a r is e s c h ie f ly fr o m t h e a s s u m p t i o n o f n o s o l id

p h a s e s i n th e c r e v i c e a n d s u g g e s ts t h a t a t s u c h l o w p o t e n t i a l s m a g n e t i t e ( F e 3 0 4 )

w o u l d b e t h e s t a b l e p h a s e . G a l v e l e i n c l u d e s t h e s o l i d p h a s e i n h i s m o d e l a n d s t i l l

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

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

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

D E T A I L E D D E S C R I P T I O N O F M O D E L O F P R O P A G A T I O N

O F C O R R O S I O N C A V I T Y

T h e p i t o r c r e v i c e is m o d e l l e d a s a p a r a l l e l - s i d e d s l o t o f l e n g t h l , w i d t h w , a n d

t h i c k n e s s d ( F i g . 1 ). T h e c r e v i c e i s a s s u m e d t o b e f il le d w i t h a d i l u t e a g g r e s s i v e

s o l u t i o n , i n th i s c a s e N a C 1 , i n i ti a ll y a t c o n c e n t r a t i o n 1 0 - 3 m o l d m -3 . T h e f o l l o w i n g

a s s u m p t i o n s a r e m a d e :

( 1 ) T h e m e t a l s u r f a c e o u t s i d e t h e c r e v i ce is a s s u m e d c o v e r e d w i th a p a s s i v e f il m

a n d t h e r e is s u f fi c ie n t g e n e r a t i o n o f c a t h o d i c c h a r g e o n t h e o u t e r s u r f a c e t o d r iv e

l o c a l i z e d c o r r o s i o n .


7/18

A model of crevice and pitting cor ros ion --I 609

( 2 ) T h e s u r f a c e a r e a o f th e s l o t is v e r y m u c h s m a l l e r t h a n t h e s u r f a c e a r e a o f th e

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

s p e c i m e n .

( 3 ) T h e t h i c k n e s s o f t h e s l o t , d is v e r y m u c h l a r g e r t h a n t h e l e n g t h , l a n d t h u s

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

( 4 ) T h e e l e c t r o l y t e is s t a t ic .

( 5) C a v i t y p r o p a g a t i o n is s lo w c o m p a r e d w i t h io n ic m i g r a t io n r a t e a n d b o t h t h e

m o v i n g b o u n d a r y a n d a n y i n d u c e d e l e c t r o l y t e m o t i o n m a y b e ig n o r e d .

( 6 ) T h e c r e v i c e s o l u t io n i s a n a e r o b i c .

( 7 ) D i l u t e s o l u t i o n t h e o r y is u s e d t h r o u g h o u t a n d t h e a c t i v it y o f w a t e r i s n o t

c o n s i d e r e d s p e c if ic a l ly .

I n o u r m o d e l , t h e f o l lo w i n g e le c t r o c h e m i c a l r e a c ti o n s a n d k i ne t ic s a r e a s s u m e d .

F o r t h e o x i d a t i o n o f i r o n

F e 2+ + 2 e - --~ F e , ( 8 )

T u r n b u l l a n d G a r d n e r 7 f o u n d t h a t b e t w e e n p H 3 a n d 8 . 5 , a n e x p r e s s io n o f t h e f o r m

il = i01 e x p

[alFE/RgT]

(9 )

w i t h i01 = 2 . 7 l 0 II A m - 2 a n d a ~ = 1 d e s c r i b e s t h e c u r r e n t q u i t e a c c u r a t e l y . E i s t h e

e l e c t r o d e p o t e n t i a l m e a s u r e d o n t h e S C E s c al e. T h e r e d u c t i o n o f w a t e r

H 2 0 + e - ~ H + O H - , ( 10 )

w a s f o u n d b y T u r n b u l l a n d T h o m a s 3 t o f o l lo w t h e r e l a t io n s h i p

i2 = i02 e x p

[a2FE/RgT]

( 1 1 )

w i t h i02 a n d a 2 i n d e p e n d e n t o f p H b e l o w v a l u e s o f 1 0 a n d t a k i n g v a l u e s - 8 1 0-1 0 A

m -2 a n d - 0 . 5 , r e s p e c t iv e l y . T h e h y d r o g e n d i s c h a r g e r e a c t io n

2 H + + 2 e - ~ H 2 , ( 1 2 )

w a s f o u n d b y T u r n b u l l a n d T h o m a s 3 t o s h o w a f irs t o r d e r d e p e n d e n c e o n h y d r o g e n

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

i 3 = i03[H +] ex p

[a3FE/RgT]

( 1 3 )

w i t h i03 = 2 1 0 - 4 A m o l e - l d m 3 m - 2 a n d a 3 = - 0 . 5 .

I n i ti a l ly t w o c h e m i c a l r e a c t i o n s a r e c o n s i d e r e d i n t h e m o d e l :

1 . F e 2+ + H 2 0 ~ - F e O H + + H + , ( 1 4 )

2 . H + + O H - ~ H 2 0 . ( 1 5 )

T h e f ir st r e a c t i o n i s s i m p l if i e d f r o m t h e m a n y - s t a g e h y d r o l y s i s t h a t is li k e l y t o o c c u r . 6

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

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

s c h e m e s c a n b e c o l l e c t e d i n t o o n e o v e r a l l s c h e m e : i t is o n l y th e e q u i l i b r i u m c o n s t a n t

w h i c h r e a l l y m a t t e r s , a n d t h is is d e t e r m i n e d b y t h e f r e e e n e r g i e s o f t h e s p e c ie s . T h u s

a n y s i m p l i f ic a t i o ns h e r e a r e l i k el y t o h a v e l es s e f f e c t o n t h e r e s u l ts t h a n t h e

s i m p l i c a t i o n s m a d e i n t h e e l e c t r o d e k i n e t ic s .

T h e e q u i l i b r i u m c o n s t a n t s o f th e r e a c t i o n s K ~ a n d K 2 c a l c u l a t e d u s i n g

RgT

In K l = G f ( F e 2+ ) + G f ( H 2 0 ) - G f ( F e O H + ) - G f ( H + )


8/18

6 1 0

S. M . SHARLAND an d P. W . TASKER

TABLE 1 . GIBB SFREE ENERGIES OF

FORMATION OF SPECIES

S p e c i e s F r e e e n e r g y k J t o o l - l

F e 2 - 7 8 . 8 7

F e O H + - 2 7 7 . 3

F e ( O H ) 2 ( s ) - 4 8 8 . 6

H 2 0 - 2 3 7 . 1 8

H 0 .0

O H - - 1 5 7 . 3

and

RgTln/ (2 = Gf(H ) + Gf(O H-) - Gf(H20 )

where G f ( S ) is the Gibbs free energy of format ion of species S, given in Table 1. The

individual forward and backward reaction rates are denoted klF, klB and k2F, k2s

where

K 1 - k l F K 2 - k 2 F

k l B ' k 2 B "

The sodium and chloride ions are assumed not to take part in the chemical reactions

but contr ibute to the transfer of current within the crevice.

In the model, it is first assumed that the crevice walls are passive, so the

steady-state mass-conservation equations for the aqueous chemical species become

o r - / 7 ]

[ dx 2 + RgT----dr. C i [- R i = 0. (16)

The six equations and the boundary conditions at the crevice tip and mouth are

specified fully in the accompanying paper.

The equations describing corrosion in two directions are derived in a similar

manner, i.e. with the electrode processes occurring on the crevice walls in addition

to the crevice tip. An approximation derived by Turnbull 8 which averages the

contributions from the walls across the width of the crevice, assuming a uniform

transverse concentration profile, is used. This effectively adds a term involving the

potential- (and hence position-) dependent currents and crevice width to the right

hand side of the mass-balance equations of the species involved in electrode

processes. The boundary conditions for this set of coupled differential equations are

as before. These equations are also given in the accompanying paper. 9

N u m e r i c a l s o l u t i o n

An analytical solution of the two sets of equations would be extremely difficult

since they are highly non-linear in nature. The non-linearity arises from the depen-

dence of the flux boundary condition at the crevice tip on the electrode potential,

which is one of the unknown variables of the system. However, by transforming to

dimensionless coordinates, rearranging and integrating, a form suitable for numeri-

cal integration may be obtained. Full details of this method are given in the

accompanying paper. 9 However, in brief, solution of the equations involved making


9/18

A m o d e l o f c r e v i c e a n d p i t t i n g c o r r o s i o n - - I 6 11

a n i n i t ia l g u e s s a t t h e p o t e n t i a l a n d p H p r o f il e s, c a l c u l a ti n g th e v a r i o u s p a r a m e t e r s

d e p e n d e n t o n t h e s e , so l vi n g t h e c o u p l e d e q u a t i o n s a n d c o m p a r i n g t h e r e s u lt a n t

p o t e n t i a l a n d p H d i s t r ib u t i o n s w i t h th e e s t im a t e s . I f t h e s e d i d n o t c o in c i d e t h e n t h e

c a l c u l a t e d p r o fi le s w e r e u s e d f o r t h e n e x t i t e r a t io n . T h i s p r o c e s s w a s c o n t i n u e d u n t il

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

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

f o r m u l a m e t h o d k n o w n a s G e a r ' s m e t h o d . S e v e r al t e c h n iq u e s i n v o lv i n g a v a r i et y o f

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

f o r s o l u t io n o f t h is s y s t e m o f e q u a t i o n s . T h e s e a r e d e s c r i b e d i n t h e n u m e r i c a l

m e t h o d s s e c ti o n o f t h e a c c o m p a n y i n g p a p e r .

C A L C U L A T E D R E S U L T S

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

p r e v i o u s l y . T h e d i f f u s io n c o e f fi c ie n t s o f a ll s p e c ie s e x c e p t H a n d O H - a r e t a k e n a s

1 1 0 - 9 m 2 s - 1 . T h e d i f f u s i o n c o e f f i c i e n t o f H i s a s s u m e d 9 . 3 x 1 0 - 9 m 2 s -1 a n d o f

O H - 5 . 3 1 0 - 9 m 2 s - 1. I n i t i a l ly a c r e v i c e o f le n g t h 2 m m a n d w i d t h 1 0 / z m i n a n e u t r a l

s o l u t i o n o f N aC 1 o f s tr e n g t h 1 0 -3 M i s c o n s i d e r e d . T h e t e m p e r a t u r e is a s s u m e d t o b e

25C

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

s o l u t io n c h e m i s t r y f o r a m e t a l p o t e n t i a l o f - 0 . 2 V . T h e p H o f t h e s o l u ti o n d e c r e a s e s

r a p i d l y n e a r t h e c r e v ic e m o u t h f r o m a b u l k v a l u e o f 7 to a b o u t 4 a n d t h e n d e c r e a s e s

s l o w l y t o a v a l u e o f 3 .4 a t t h e c r e v i c e t ip . L a r g e q u a n t i t i e s o f F e 2 a r e g e n e r a t e d w i t h

c o n c e n t r a t i o n s r e a c h i n g m o r e t h a n 1 0 M a t t h e c r a c k ti p . ( A t s u c h h i g h c o n c e n t r a -

t io n s t h e a s s u m p t i o n o f d i l u t e s o lu t i o n t h e o r y is n o t v a l i d .) 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 ls o i n c re a s e s r a p i d l y w i th d i s t a n c e a l o n g t h e c r e v i c e e n s u r i n g c h a r g e

n e u t r a l i t y is m a i n t a i n e d e v e r y w h e r e . F i g u r e 3 sh o w s t h e c h e m i s t r y f o r a m e t a l a t

p o t e n t i a l 0 . 0 V . T h e [ C I - ] a n d [ F e 2 ] r is e a n d p H d r o p a l o n g t h e p i t a r e m o r e

s ig n i fi c a n t f o r h i g h e r p o t e n t i a l s . F i g u r e 4 s h o w s a c o m p a r i s o n o f t h e p o t e n t i a l d r o p s

21 Ct-

~r 0

F e z

~ .

c

~ -2 FeOH*

. . . . . . . . . . . . . . .

o -/

o

-6

N o *

_~

0 3 0 6 0-9 12 1'5 1.8 2 0

Dis tonce f rom eev i ty t ip , ram

I

1 i

I

I

I

~bM =-O '2v , l=2 m m , [ C l '18ulk=10 3 M

FIG . 2. C o n c en t r a t i o n p r o f i l e s a l o n g t h e cav i t y l en g t h f o r a c rev i ce w i t h p as s i v e w a l l s.


10/18

6 1 2 S .M . SHA RL AN D an d P .W . T ASKE R

2 . 5

:C 0

o

:,7-

2 .s

o

u

- 5 . 0

o

o

- 7 ' 5

- 1 0 ' 0

FIG. 3.

C I

F e O H 4

I

/ '

J

N 0 ~ ' ~ ' ~ '

l

0 3 0 6 0 ' 9 1 2 1 5 1 8 2 ' 0

D i s t a n c e f r o m c a v i t y t i p . t u r n

~ M = O ' O V , I = 2 r a m . [ E l ' ] B u l k = 1 0 3 M

C o n c e n t r a t i o n p r o f il e s a l o n g t h e c a v i ty l e n g t h f o r a c r e v i c e w i t h p a s s i v e w a l ls .

a l o n g t h e c av i t y l e n g th f o r m e t a l s a t p o t e n t i a l s - 0 . 4 , - 0 . 2 a n d 0 . 0 V . T h e c a l c u l a t e d

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

u n d e r t h e s e t h e o r e t i c a l c o n d i t i o n s . I t w ill b e s h o w n t h a t i n c l u s io n o f c o r ro s i o n o n t h e

c a v i t y w a l l s r e d u c e s t h e c o r r o s i o n c u r r e n t d e n s i t ie s c o n s i d e r a b l y . F i g u r e 5 s h o w s t h e

v a r i a ti o n o f c o r r o s i o n c u r r e n t w i t h c a v i t y l e n g t h ( w i th e a c h c a l c u l a t i o n c a r r ie d o u t

i n d e p e n d e n t l y i n a s t at ic g e o m e t r y , i .e . n o m o v i n g b o u n d a r i e s a r e i n v o l v e d h e r e ) f o r

t h e c a s e q~M = - 0 . 2 V o A l t h o u g h t h e m a g n i t u d e s o f t h e c u r r e n t s a r e p h y s i c a ll y

u n r e a l i s ti c ( f o r r e a s o n s w h i c h w il l b e s u g g e s t e d l a t e r ) , t h e g e n e r a l t r e n d o f d e c r e a s i n g

cur r en t wi th inc r eas ing d i f fus ion l eng th i s s ign i f i can t .

> 0 / , ~

-

0 3

0.

o 0 2

' 6 0 - 1 -

c

e

o

Q.

0

0

I I I I . . . . . I I

0 ' 3 0 6 0 9 1 2 1 5 1 ' 8 2 0

D i s t a n c e f r o m c a v i t y t i p , ra m

I = 2 m m . [ C l ' ]B u t k = 1 0 3 M

FIG . 4. E l ec t r o s t a t i c p o t en t i a l d r o p a l o n g cav i t y l en g t h f o r a c rev i ce w i t h p as s i v e w a l l s.


11/18

A m o d e l o f c r e v i c e a n d p i t t i n g c o r r o s i o n - - I 6 1 3

%

x

L

0

10

g

8

7 -

6 -

5

3 -

2 L

I I i I I

2 . 0 4 0 6 - 0 8 .0 1 0 . 0

C a v i ty l e n g t h . m m

~ M = 0 . 2 V . I C l ] B u t k = 1 0 3

P a s s i v e w a l l s

FIG . 5 , C o r r o s i o n cu r r e n t ag a i n s t cav i t y l en g t h .

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

H o w e v e r , t h e s e c u r r e n t s a r e v e r y sm a l l a t s u c h a n o d i c e l e c t r o d e p o t e n t i a l s a n d it is

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

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

F i g u r e 6 sh o w s t h e s o l u t i o n c h e m i s t r y in a c a v i ty w i t h c o r r o d i n g w a l ls a n d b a s e , a t

a p o t e n t i a l o f - 0 . 2 V . T h e s a m e b a s ic t r e n d s a s f o r th e p a s s iv e w a ll c a s e a r e o b s e r v e d ,

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

:E

5

c

o

o

2 5

5 C

7 5

I 0 C

C I -

2 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

F e 2 . L . . . . ~ \

0 - t

F e O H * I

- H

t

I I I I I I I

0 3 0 - 6 0 '9 1 2 1 5 1 8 2 0

D i s t a n c e f r o m c a v i t y t i p , r a m

q~M = - 0 2 V ' [ = 2 m m ' [ C l - ] e u l k = 1 0 -3 M

FIG . 6 . C o n c en t r a t i o n p r o f i l e s a l o n g t h e cav i t y l en g t h f o r a c r ev i ce w i t h co r r o d i n g w a l ls .


12/18

614 S .M . SHARLAND nd P. W. TASKER

0.35 I

0 30

o z 5

o, 0 20

c

_o

o

, , 0 15

1o

0 ' 1 0

_

o

a. 0.05

0

Fro. 7.

A c t i v e

w a l l s

I I I I

0.3 0.6 0.9 1.2 1-5 1.8 2 0

D i s ta n c e f r o m c a v i ty t i p . r a m

~M = 0 '2 V . l=2 m m , [ C I - ]Bu lk = 10 -3 H

Comparison of the potential drops along the crevice length for corroding and

non-corroding walls.

t h e c a v i t y t ip . T h e m a g n i t u d e s o f th e d e v i a t io n s f r o m b u l k v a l u e s , h o w e v e r , a r e

s l ig h t l y l a r g e r w i t h c o r r o d i n g w a l ls . F o r e x a m p l e , t h e p H v a l u e a t t h e c a v i ty ti p is

a b o u t 3 c o m p a r e d w i t h 3 . 4 fo r t h e p a s s iv e w a ll s. T h e p r o f il e s a ls o s h o w m o r e r a p i d

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

p a s s iv e w a l l s i n F i g . 7 i n d i c a t e s t h a t t h e c o r r o s i o n c u r r e n t s a r e r e d u c e d w h e n m e t a l

d i s s o l u t i o n o c c u r s o v e r a la r g e r a r e a . I n t h i s c a s e t h e c o r r o s i o n c u r r e n t a t t h e p i t b a s e

i s a p p r o x i m a t e l y 2 x 1 0 2 A m - 2 c o m p a r e d w i t h 3 x 1 03 A m - 2 f o r t h e p a s s i v e w a l l s .

T h e s e c u r r e n t s a r e s ti ll p h y s i c a l ly u n r e a l i s ti c a n d a n u m b e r o f o t h e r p h e n o m e n a n e e d

t o b e c o n s i d e r e d . O n e i m p o r t a n t a d d i t i o n i s t h e p r e c i p i t a t io n o f s o li d f e r r o u s

h y d r o x i d e , s i n c e i n a ll t h e e x a m p l e s g i v e n t h e s o l u b i l i ty li m i t is e x c e e d e d . I t w i ll b e

s h o w n t h a t c o n s t r ai n in g t h e r a ti o o f p H v a l u e t o F e O H re d u c e s t h e c o r r o s i o n

c u r r e n t s s l ig h t ly a n d s c a li n g d o w n t h e d i f f u s io n c o e f f ic i e n ts t o s i m u l a t e t h e p r e s e n c e

o f s o li d h y d r o x i d e r e d u c e s t h e m f u r t h e r t o p h y s i c a ll y r e a li s ti c v a l u e s.

assivation of crevice walls

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

d i s t r ib u t i o n o v e r t h e i n t e r n a l m e t a l s u r f ac e . F i g u r e 8 s h o w s t h e c o r r o s i o n c u r r e n t a s

a f u n c t i o n o f p o s it i o n in a ca v i t y o f le n g t h 2 m m a t a p o t e n t i a l - 0 . 2 V . T h e c u r r e n t a t

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

p a s s i v a t i o n o f t h e m e t a l . I n r e a l i t y , it s e e m s l i k e l y t h a t t h i s r e g i o n w o u l d r e p a s s i v a t e 1

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

A l t h o u g h t h e p r e d i c t e d c u r r e n t s i n t h e r e s t o f t h e c a v i ty a re u n r e a l is t ic a l ly h ig h t h e

d i s t r ib u t i o n g i v e s a n i n d i c a t i o n o f t h e f u t u r e s h a p e o f th e p i t ; t h e r a p i d b r o a d e n i n g o f

t h e w i d t h c lo s e to t h e m o u t h w i th a m o r e e v e n b r o a d e n i n g o v e r t h e m a j o r i t y o f th e

l e n g t h s u g g e s t s a n e v e n t u a l b o t t l e - s h a p e d c a v i ty . M o s t o f t h e c u r r e n t d i s t ri b u t i o n s

g e n e r a t e d f o r th e r a n g e o f p a r a m e t e r s i n t h is p a p e r d i s p la y th i s s t u d i e d b e h a v i o u r .


13/18

A model of crevice and pitting corro sion --I 615

6 5

t~

' 6 C

E

c 5 ' 5

g

5 0

~ / .5 -

o

u

~ 4 0-

o

3 'C

Fl~. 8.

I

0 3

f

~ ' t - - - ' - - - - ~ ~ I I I

0 '6 0 9 1 '2 1 5 1 8 2 -0

D i s ta n c e f r o m c a v i t y t i p , r a m

~ M = ' 0 2 V , I = 2 m m . [ C L '] B u t k = 2 x 10 2 1.1

Corrosion current against cavity length for a crevice with active walls.

P R E C IP I T T I O N O F F E R R O U S H Y D R O X I D E

I n a ll t h e e x a m p l e s g i v e n in e a r li e r s e c ti o n s t h e p r o d u c t s o f f e r r o u s a n d h y d r o x i d e

i o n s h a v e e x c e e d e d t h e s o l u b i l i ty li m i t o f so l id f e r r o u s h y d r o x i d e .

s a

n e x t s t a g e i n

t h e m o d e l t h e r e f o r e , t h e r e a c t io n

F e O H + + H 2 0 ~ F e ( O H ) 2 + H ( 17 )

is c o n s i d e r e d s p e c if ic a l ly . T h e e q u i l i b r i u m c o n s t a n t o f th i s r e a c t i o n is c a l c u l a t e d f r o m

t h e f r e e e n e r g i e s o f t h e i n d iv i d u a l sp e c i es a s b e f o r e . I n a r e c e n t p a p e r , G r a v a n o a n d

G a l v e l e 6 t r e a t t h e s o l i d a s a d i f fu s i n g s p e ci e s . T h e y i n c l u d e a t e r m D dC/dx)d i r e c t l y

i n t h e i r m a s s - b a l a n c e e q u a t i o n s a n d s o l v e f o r th i s s o li d 'c o n c e n t r a t i o n ' , C i n t h e s a m e

w a y a s f o r t h e o t h e r c o n c e n t r a t i o n s . T h e d i f f u s i o n c o e f f i c i e n t , D i s s e t t o 1 1 0 - 9

m E s , t h e s a m e a s t h e o t h e r d i f f u s i n g i o n s ( e x c e p t H a n d O H - ) . T h i s d e s c r i b e s t h e

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

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

s ta g e s o f t h e p r o c e s s . T h e m o d e l d e s c r i b e d h e r e c o n s i d e rs t h e e f f e c ts o f p r e c ip i t a t io n

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

c o n c e n t r a t i o n a n d h e n c e a v o i d s t h e n e c e s s i t y to a s s ig n a d if f u s io n c o e f f i c ie n t t o a

p o t e n t i a l l y n o n - m o b i l e s p e c i e s .

T h e r e i s l it tl e k i n e t ic d a t a a v a i l a b l e o n t h e r a t e o f p r e c i p i t a t i o n . G a l v e l e 5 s u g g e s t s

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

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

m e t a l - s o l u t i o n i n t e r f a c e t h e p r o c e s s w i l l b e m u c h f a s t e r . F o r s i m p l i c i t y , i t w i l l b e

a s s u m e d in th i s m o d e l t h a t t h e p r e c i p i t a t io n r a t e is f a s t e v e r y w h e r e c o m p a r e d t o t h e

d i f f u si o n r a t e s a n d t h e m a s s - b a l a n c e e q u a t i o n s w i ll b e m o d i f i e d s o t h a t t h e s o l u b i li t y

o f F e ( O H ) 2 is n o t e x c e e d e d .

T h e g o v e r n i n g m a s s - t r a n s p o r t e q u a t i o n s a r e d e r iv e d a s b e f o r e b u t w i t h a n


14/18

616 S .M . SHARLAND an d P. W. TASTER

d

_

-

8

- / .

9

Fro. 9.

I

1 I I I I I

0 6 0 ' 9 1 2 1 5 1 8 2 0

D i st an c e f ro m c o v i t y t i p , m m

C I

2

F e z ~ ~ .

0 FeO H+ .....................................

- 6

: . _ . N o

. . . . . ~ - ~ ~ ' ~ '~ x

-~

0 0.3

~b~4

=

-0"2

V, I

=2ram. [CI" Built=

2

x l 0 " 2

M

C o n c e n t r a t i o n p r o f i le s a l o n g t h e c a v i t y l e n g t h f o r a c r e v i c e w i t h p a s s i v e w a ll s ( w i t h

a p r e c i p i t a t i o n r e a c t i o n i n c l u d e d b u t n o c h a n g e t o d i f f u s io n c o e ff i c ie n t s ).

additional constraint fixing the ratio of the H and FeOH + concentrations. The

boundary conditions at the pit mouth are modified slightly to ensure both charge

neutra lity and chemical equilibrium here. The details of the modified equations and

boundary conditions may be found in the accompanying paper. The equations are

solved using the iteration techniques employed earlier. Figure 9 shows the predicted

solution chemistry in a pit with non-corroding walls. The p r meters are the same as

in the previous calculations except that the bulk chloride concentration is assumed 2

10 -2 M. The diffusion coefficients are assumed 10 -9 m 2 s -1 (except those ofH + and

OH -) for this calculation. Compar ison of Fig. 9 with a run with the solubility limits

exceeded shows a much lesser degree of acidification, about pH 5 at the cavity tip

compared with pH 3 (and a correspondingly lower [FeOH+]). However, the potential

distributions in the crevices are very similar and the corrosion currents are only

reduced by a few per cent by constraining the pH and FeOH + in this way. Figure 10

shows the results of the 'active-wall' calculation using the same parameters as in

Fig. 9. Again, it may be seen that the solution is more acidic when the crevice walls

are corroding in addition to the base.

In all the examples given the predicted rates of metal dissolution il, are very large.

However, no account has been taken of the physical effect of the corrosion product

restricting the migration of aqueous species. There are little available empirical data

on the diffusion coefficients in porous iron solids, so the model is tested over a range

of values. Figure 11 shows the variation of the corrosion currents with diffusion

coefficients for the case of an active wall cavity. The reduction in the corrosion

current is quite significant as the diffusion of both aggressive species and corrosion

products becomes impaired. Figure 12 shows the solution chemistry with a diffusion

coefficient of 1 0 - 1 3 m 2 s -1. Comparison with Fig. 10 shows a higher degree of

acidification is obtained since the hydrogen ions produced in the hydrolysis reaction

cannot diffuse out of the pit as easily.


15/18

A m o d e l o f c r e v ic e a n d p i t ti n g c o r r o s i o n - - I 6 17

2.5

0

~ E

d

o

"6

, - - 2 E

E

tJ

- 5 ' (

o~

-7 -5

-10"0

Fx6. 10.

C l -

F e 2 . . . . . . . . . . . .

F e O H *

H

N a

J

I I I t I I

0.3 0.6 0.9 1-2 1.5 1.8

D i s ta n c e f r o m c a v i t y t i p , ra m

q b N = - 0 2 V , l = 2 m m , [ C l - ] B u l k = 2 x 1 0 - 2 N

2 0

C o n c e n t r a t i o n p r o f i l e s a l o n g t h e c a v i t y l e n g t h f o r a c r e v i c e w i t h a c t iv e w a l l s ( w i th

a p r e c i p i t a t i o n r e a c t i o n b u t n o c h a n g e t o d i f f u s i o n c o e f fi c i e n t s ).

E

x

E

L

o

t~

o

L

o

9 C

8'C

7 0

6 0

5 0

4 0

3 0

2 0

1 0

I I I I ~ ~ - ~ - - T ~ I

0 9 0 9 5 10 .0 10 '5 11 .0 11 .5 12"0 12 '5

- I o g l o d i f f u s i o n oeffi ient o f F e z . e t c . , m z s - |

~ M = _ 0 . 2 V , I = 2 m m , [ C t ' ] B u l k = 2 x 1 0 -2 I~t

1

1s.O

F IG . l l . C o r r o s i o n c u r r e n t a g a i n s t d i f f u s i o n c o e f f i c i e n t o f F e 2 , F e O H + , N a + a n d C I f o r

a c r e v i c e w i t h a c t i v e w a l ls


16/18

618 S .M . SHARLAND an d P. W . TASKER

5"0

.,~ 2'5

~

C

c - 2 " 5

-5.o

.7.~

-10 (

Fro. 12.

C[-

F e Z . . . . . . . . .

FeOH *

H

Na* .~ . ----..-

, ' - -

~ i ' - - I - - J ' - - I I I I

0 0-3 0.6 0 -9 1.2 1.5 1.8 2.0

Distance from cavi ty tip, ram

~M =-0.2V,I= 2mm,[Cl ]Butk = 2xI0 2 M ,D =I0 -13 m2 s -I

C o n c e n t r a t i o n p r o f i le s a l o n g c a v i t y l e n g t h f o r a c r e v ic e w i t h a c t i v e w a ll s ( p r e c i p i-

t a t i o n r e a c t i o n i n c l u d e d a n d d i f f u si o n c o e f fi c ie n t s r e d u c e d ) .

L I M I T A T I O N S A N D F U T U R E E X T E N S I O N S T O T H E M O D E L

The steady-state model described in this paper gives some reasonable qualitative

predictions of the chemistry and electrochemistry within the corroding crevice but

the quantitative comparisons with empirical cavity propagation rates and ionic

concentrations are less encouragingl The inaccurate cavity propagation rates pre-

dicted, however, seem consistent with other models in the literature. This would

suggest that the approximations made in the construction of the model are not

strictly valid and that physical processes have been oversimplified or the empirical

input data is not sufficiently accurate. In many of the examples, the predicted

solutions are highly supersa turated with respect to ferrous chloride. The addition of

the equilibria reaction

Fe 2+ + 2C1- ~ FeCI2 (18)

would seem necessary in these cases. However, limitations with the numerical

solving method of the mass-conservation equations restrict the addition of further

chemical equilibria reactions. There is therefore a need to develop an alternative

solving technique in the model to include a more complex description of the

chemistry within the crevice.

In most of the examples given, the effect of the moving boundary is negligible and

the approximation of a static geometry is valid for a steady-state calculation.

However, at a high potential, e.g. 0.0 V in pits with passive walls, the calculations

produce currents of, the order of 105 A m -2 which is roughly equivalent to the metal

interface retreating at a speed of 1 x 10 -2 mm s -1. This is fast compared with the

diffusive speed of the mobile species and consequently the moving boundary and the

induced electrolyte motion should be specifically considered for such sets of par-

ameters.


17/18

A m odel of crevice and pitting corrosion--I 619

A n o t h e r p o s si b le e x t e n s io n o f t h e m o d e l m a y b e a c o n s i d e r a t i o n o f t h e e f f e ct o f

l im i t i n g t h e c a t h o d i c c u r r e n t g e n e r a t e d o n t h e e x t e r n a l s u r f a c e . A l s o t h e e l e c tr i ca l

r e s i st a n c e o f t h e b u l k s o l u t i o n , w h i c h m o d i f ie s t h e c h a r g e t r a n s f e r a n d m a y l im i t t h e

c o r r o s i o n r a t e s , s h o u l d b e t a k e n i n t o a c c o u n t .

C O N C L U S I O N

A m o d e l o f th e p r o p a g a t i o n s t ag e o f c r e v ic e o r p i tt i n g c o r r o s i o n w h i c h is e n t i r e l y

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

s o d i u m c h l o ri d e s o l u t i o n . T h e s t e a d y - st a te e q u a t io n s g o v e r n i n g m a s s t r a n s p o r t a n d

e l e c t r o d e p o t e n t i a l o f a s y s t e m o f m o b i l e s p e c i es i n a r e s t ri c t e d g e o m e t r y a r e s o l v e d

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

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

s e p a r a t e l y a c ti v e c o r r o s i o n a t th e c r e v i c e b a s e o n l y a n d c o r r o s i o n o n t h e b a s e a n d

w a l ls . A n e x t e n s i o n t o th e m o d e l i n c l u d e s a n a c c o u n t o f th e e f f e c t o f p r e c i p it a t io n o f

s o li d fe r r o u s h y d r o x i d e . T h e m a i n c o n c l u s i o n s a r e a s f o ll o w s :

( 1) F o r b o t h a c ti v e a n d p a s s iv e w a l l s, t h e r e s u lt s s h o w d e c r e a s i n g p H v a l u e s a n d

i n c r e a si n g F e 2 a n d C I - c o n c e n t r a t i o n s a l o n g t h e p i t t o w a r d s t h e t ip , w i th d e v i a t io n s

o f i n d i v i d u a l sp e c i es f r o m t h e i r b u l k v a l u e s i n c re a s i n g w i t h in c r e a s in g m e t a l p o t e n -

t ia l. B y v a r y i n g th e m a n y p a r a m e t e r s o f t h e s y s t e m it w o u l d b e p o s s i b le t o i d e n t i f y

c r i t i c a l p h y s i c a l c o n d i t i o n s ( e . g . b u l k p H , b u l k C 1 - c o n c e n t r a t i o n ) w h i c h r e s u l t i n

p i t t i n g o r c r e v i c e c o r r o s i o n .

( 2 ) I n t h e s i m p l e s t m o d e l , w i t h n o s o l i d c o r r o s i o n p r o d u c t s , t h e c a l c u l a t e d

c o r r o s i o n c u r r e n t s f o r t h e p a s s i v e w a l l c a s e a re v e r y h ig h ( f o r CM = - 0 . 2 V , l = 2 m m ,

i = 3 x 103 A m - 2 ) b u t a r e r e d u c e d b y t h e i n c l u s i o n o f c o r r o d i n g w a l l s (i --- 2 x 102 A

m - 2 ) . W h e n a c c o u n t is t a k e n o f th e e f f e c t o f a s o li d p r e c i p i ta t e ( f e r r o u s h y d r o x i d e )

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

t h e s e v a l u e s a r e u n r e a l is t ic a l ly h ig h w h i c h w o u l d s u g g e s t t h a t s o m e o f t h e a p p r o x i -

m a t i o n s m a d e i n t h e c o n s t r u c t i o n o f t h e m o d e l m a y n o t b e v a li d . T h e d e s c r i p t io n o f

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

t r e a t m e n t o f th e s o l id p r e c ip i t a te s . F o r e x a m p l e , t h e p r e d i c t e d s t e a d y - s t a t e so l u t io n s

w i t h i n t h e c r e v ic e a r e h i g h l y s u p e r s a t u r a t e d w i t h re s p e c t t o f e r ro u s c h l o r i d e . F u r t h e r

m o d e l d e v e l o p m e n t is n e c e s s a r y t o i n v e s t i g a t e th i s ef f e c t.

( 3) T h e s o li d c o r r o s i o n p r o d u c t s h a v e a n e x t r e m e l y i m p o r t a n t e f f e c t o n t h e c a v i t y

p r o p a g a t i o n r a te s t h r o u g h t h e i r l i m i ti n g e ff e c t o n t h e d i f f u s i o n io n i c sp e c i e s w i t h i n

t h e c r e v i c e . T h e r e i s l it tl e a v a i l a b l e e m p i r i c a l d a t a o n d i f f u s i o n c o e f f ic i e n t s w i t h i n

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

t o b e a c c u r a t e l y d e t e r m i n e d i f t h e m o d e l is to b e u s e d i n a p r e d i c ti v e r o l e in t h e f u t u r e .

( 4) F o r t h e r a n g e o f m e t a l p o t e n t ia l s c o n s i d e r e d ( - 0 . 4 4 ) . 0 V ) t h e s i g n if ic a n t p a r t

o f c a t h o d i c a c t iv i t y o c c u r s o n t h e m e t a l s u r f a c e o u t s id e t h e c r e v ic e m o u t h .

( 5 ) B y c o n s i d e r i n g t h e p r o p o r t i o n o f t h e c r e v i c e l e n g t h a t w h i c h t h e p o t e n t i a l

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

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

e s t i m a t e t h e e v e n t u a l s h a p e o f th e c r e v ic e . T h e m o d e l p r e d i c t s a ra p i d b r o a d e n i n g o f

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

o v e r t h e m a j o r i t y o f t h e l e n g t h . T h i s w o u l d s u g g e s t a n e v e n t u a l ' b o t t l e - s h a p e d '

c r e v i c e .

T h e m o d e l w i ll b e d e v e l o p e d f u r t h e r t o i n c l u d e a b e t t e r d e s c r i p ti o n o f th e s o l u t i o n

c h e m i s t r y in t h e c r e v i c e .


18/18

620 S .M . SHARLAND an d P. W . TASKER

A c k n o w l e d g e m e n t s - - T h i s

w o r k w a s j o in t l y f u n d e d b y t h e D e p a r t m e n t o f t h e E n v i r o n m e n t ( D O E ) a s p a r t

o f i t s r a d i o a c t i v e w a s t e m a n a g e m e n t r e s e a r c h p r o g r a m m e , a n d t h e C o m m i s s i o n o f t h e E u r o p e a n

C o m m u n i t i e s . I n t h e D o E c o n t e x t , t h e r e s u l ts w il l b e u s e d i n t h e f o r m u l a t io n o f G o v e r n m e n t p o l ic y , b u t

a t t h i s s t a g e d o n o t n e c e s s a r i l y r e p r e s e n t G o v e r n m e n t p o li c y .

R E F E R E N C E S

I. G . P. MARSH , K. J . TAYLOR, I . D . BLAN D, C. WESTCOTT, P. W . TASKER an d S . M . SHARLAND,Proc .

M R S S y r u p . S c i e nt if ic B a s i s o r N u c l e a r W a s t e M a n a g e m e n t S t o c k h o l m , S w e d e n ( E d i t e d b y L . W ER N E,

p . 421) (Sep t . 1985).

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3 . A . T U RN BU LL an d J . G . N . T HO M AS, N P L D M A R ep o r t ( A ) , 2 3 ( 1 9 8 0 ).

4. J . R. GALVELE,

J . E l e c t roc he m . Soc .

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6. S. M . GRAVANOan d J . R . GALVELE,

Corros . Sc i .

2 4 ,5 1 7 ( 1 9 8 4 ) .

7. A . TURNBULL,Br. Co rros . J . 15, 162 (1980) .

8. A . TURNBtJLLand M. R. GARDNER, Corros . Sc i . 2 2 ,6 6 1 ( 1 9 8 2 ) .

9. S. M. SHARLAND,

Corros . Sc i .

2 8 ,6 2 1 ( 1 9 8 8 ) .