multiphase diffusion in binary and ternary solid-state systems · multiphase diffusion in binary...

Multiphase diffusion in binary and ternary solid-statesystemsvan Loo, F.J.J.

Published in:Progress in Solid State Chemistry

DOI:10.1016/0079-6786(90)90007-3

Published: 01/01/1990

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Peoq. Solid St. Chem. Vol. 21). pp. 47-99. 19~1 (MI79--6786/t~I SI).OO ~e .5U Printed in Or~at Britain. All rights re. f red ~) I ~ ) Pergamon Press plc

MULTIPHASE DIFFUSION IN BINARY AND TERNARY SOLID-STATE SYSTEMS

F. J. J. van Loo

Laboratory for Solid-State Chemistry and Materials, Eindhoven University of Technology, Eindhoven, The Netherlands

1. INTRODUCTION

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

w i l l o c c u r . Depending on the n a t u r e of the s t a r t i n g m a t e r i a l s , t e m p e r a t u r e and a n n e a l i n g

t ime , a new 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 of t he e l e m e n t s i s s e t up. I f the m a t e r i a l s a re

c o m p l e t e l y m i s c i b l e a t t h a t t e m p e r a t u r e , the c o n c e n t r a t i o n p r o f i l e s a r e more or l e s s smooth

w i t h o u t any d i s c o n t i n u i t y . I f , however , they a r e on ly p a r t i a l l y m i s c i b l e or i f t hey r e a c t to

form new p h a s e s , d i s c o n t i n u i t i e s i n the c o n c e n t r a t i o n p r o f i l e s occur which a re c l o s e l y

r e l a t e d to t he phase d iagram of t he sys tem.

T To

(a)

m

c

I(b)

/ /

II II I I ' ! ' I ' mole fraction

ct 11 I I / y ~ _ _ J T - -

8 IB

C

A X

Fig. i.i.

Fig. 1.2.

(a ) Phase d iagram of t he b i n a r y sys tem A-B. (b) A d i f f u s i o n coup le A/B annea l ed a t a t e m p e r a t u r e T o , t o g e t h e r w i t h the c o n c e n t r a t i o n - p e n e t r a t i o n curve of component B in mole f r a c t i o n . Hypothetical ternary isotherm A-B-C with two possible diffusion paths, 1 and 2.

47

4~ F.J. J, van Loo

F i g u r e I . I shows an example f o r a b i n a r y sys tem A-B, i n which t h e compounds ¥ and 6 a r e

formed i n a d d i t i o n to t h e s o l i d s o l u t i o n s m and ~. A f t e r a s u f f i c i e n t l y long a n n e a l i n g

t r e a t m e n t f o r such an A-B d i f f u s i o n c o u p l e a t a t e m p e r a t u r e T o, a c o n c e n t r a t i o n p r o f i l e i s

s e t up as shown f o r component B. The e x a c t p r o f i l e i n t h e v a r i o u s p h a s e s c a n n o t be p r e d i c t e d

u n l e s s t he i n t e r d i f f u s i o n c o e f f i c i e n t s a r e known as a f u n c t i o n of c o m p o s i t i o n . However, t he

v a l u e s of t he c o n c e n t r a t i o n s a t t h e m/y , y / 6 and ~/B i n t e r f a c e s can be t a k e n from the phase

d iagram i f e q u i l i b r i u m c o n d i t i o n s p r e v a i l a t t h e s e i n t e r f a c e s ( a s i s d i s c u s s e d l a t e r ) .

The r e a s o n f o r t h e c r e a t i o n of s t r a i g h t i n t e r f a c e s w i t h f i x e d c o n c e n t r a t i o n gaps i n

b i n a r y m u l t i p h a s e d i f f u s i o n c o u p l e s f o l l o w s from b a s i c thermodynamic c o n s i d e r a t i o n s . From

the phase r u l e i t f o l l o w s t h a t o n l y s i n g l e - p h a s e r e g i o n s can be formed i n such a c o u p l e ,

b e c a u s e t h r e e d e g r e e s of f reedom would be n e c e s s a r y to f i x t e m p e r a t u r e and p r e s s u r e and to

v a r y t he c o n c e n t r a t i o n . Two-phase r e g i o n s ( i . e . p r e c i p i t a t e s o r wavy i n t e r f a c e s ) a r e ,

t h e r e f o r e , f o r b i d d e n s i n c e t h e n o n l y two d e g r e e s of f reedom a r e a l l o w e d . Tempera tu re and

p r e s s u r e a l r e a d y h a v i n g been c h o s e n , t h e r e would be no p o s s i b i l i t y l e f t a c r o s s a two-phase

zone for a variation in concentration, i.e. for diffusion.

For Interdlffusion in a ternary system at a fixed temperature and pressure, the number of

degrees of freedom is three minus the number of phases. This implies that in slngle-phase

regions one might vary the concentration (fraction) of two components, whereas in two-phase

regions the concentration of one component can be varied but then the other concentrations

are fixed. Therefore, in a ternary diffusion couple two-phase regions are allowed.

In a sys tem where a component i s i n v o l v e d which i s p r e s e n t b o t h i n t h e s o l i d and i n the

s u r r o u n d i n g gaseous phase i t i s n o t e a s y to f i x unambiguous ly p a r a m e t e r s l i k e t o t a l p r e s s u r e

or t he f r a c t i o n of t h i s component i n t he s o l i d p h a s e , s i n c e t h e s e may be d i f f e r e n t i n t he

b u l k of the c o u p l e and a t t he edges as w i l l be e x p l a i n e d l a t e r on.

A l though n a t u r e o n l y chooses one r e p r o d u c i b l e p e n e t r a t i o n p r o f i l e f o r t he e l e m e n t s i n

such a t e r n a r y c o u p l e , a p r e d i c t i o n of t h e s e p r o f i l e s f rom t h e p h a s e d i ag ram i s d i f f i c u l t

b e c a u s e of t h e i n f i n i t e v a r i e t y of h y p o t h e t i c a l p o s s i b i l i t i e s . In F i g u r e 1 .2 an i s o t h e r m a l

c r o s s - s e c t i o n of a sys tem A-B-C i8 shown, e x h i b i t i n g two s i n g l e - p h a s e f i e l d s y and S and a

two-phase region in which tie lines are drawn connecting coexisting equilibrium phases.

If the element C and the alloy X Interdlffuse at the temperature for this cross-section,

the resulting concentration profile mapped on this section might resemble the hypothetical

d i f f u s i o n p a t h s 1 or 2 o r any o t h e r , as l ong as mass b a l a n c e i s p r e s e r v e d . The

c o n c e n t r a t i o n s a t t h e y - 6 i n t e r f a c e and t h e morphology of t he r e a c t i o n zone would be

c o m p l e t e l y d i f f e r e n t f o r p a t h s 1 and 2 a s i s d i s c u s s e d l a t e r .

In F i g s . 1 .1 and 1 . 2 , we have r e l a t e d t h e d i f f u s i o n phenomena w i t h t he r e l e v a n t phase

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

c a n n o t be c o n s i d e r e d g e n e r a l l y . For v e r y s h o r t a n n e a l i n g t i m e s o r i n t h i n - f i l m e x p e r i m e n t s ,

d e p a r t u r e from e q u i l i b r i u m d iagram c o n c e n t r a t i o n s c e r t a i n l y o c c u r b e c a u s e of s u r f a c e e n e r g y

c o n t r i b u t i o n s , b u t a l s o f o r l o n g e r d i f f u s i o n a o n e a l i n g t i m e s , d e v i a t i o n s from e q u i l i b r i m n

c o n d i t i o n s a t t h e phase i n t e r f a c e s a r e r e p o r t e d [ 1 - 3 ] . The r e l a t i o n be tween t h e d i f f u s i o n

phenomena and t h e phase d iagram i s t h e n l e s s o b v i o u s , a l t h o u g h i n some c a s e s m e t a s t a b l e

e x t e n s i o n s of e q u i l i b r i u m l i n e s m i g h t be h e l p f u l i n u n d e r s t a n d i n g t h e s e d e v i a t i o n s . In some

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

Local e q u i l i b r i u m can a lways be assumed i f volume d i f f u s i o n i s t he r a t e - l i m i t i n g s t e p .

Whether t h i s i s t he ca se can be v e r i f i e d by p l o t t i n g t h e d i f f u s i o n p r o f i l e s f o r ( a t l e a s t )

two s u f f i c i e n t l y d i f f e r e n t a n n e a l i n g t i m e s t a s a f u n c t i o n of x / t ~, where x i s the d i s t a n c e

p a r a m e t e r . In c a s e of a d i f f u s i o n - c o n t r o l l e d p r o c e s s , t h e p l o t s t h e n have to c o i n c i d e .

Multiphase Diffusion 49

In t h i s s u r v e y t h e a u t h o r would l i k e to d i s c u s s i n a m a i n l y q u a l i t a t i v e way t he r e l a t i o n

be tween m u l t i p h a s e d i f f u s i o n phenomena on t h e one hand , and k i n e t i c and thermodynamic (phase

d i ag ram) d a t a on t h e o t h e r hand . Th i s p a p e r t h e n f o c u s e s on:

- p r o v i d i n g e x p e r i m e n t a l o r t h e o r e t i c a l e v i d e n c e to s e t t l e some o f t e n d i s p u t e d i s s u e s ;

- showing how r e l a t i v e l y s i m p l e e x p e r i m e n t s can p r o v i d e i n f o r m a t i o n a b o u t d i f f u s i o n and

the rmodynamics ;

- p r e d i c t i n g and e x p e r i m e n t a l l y v e r i f y i n g m u l t i p h a s e d i f f u s i o n phenomena from k i n e t i c and

thermodynamic d a t a .

This p a p e r i s b a s e d m a i n l y on t h e a u t h o r ' s own e x p e r i e n c e s i n t h i s r e s e a r c h f i e l d and i s

n o t meant a s a c o m p r e h e u s i y e l i t e r a t u r e s u r v e y o r t o t a l o v e r v i e w . For i n s t a n c e , n o t o r o n l y

p a r t l y i n c l u d e d i n t h i s p a p e r a r e :

- N o n - e q u i l i b r i u m d i f f u s i o n c o u p l e s , a s i t u a t i o n o f t e n e n c o u n t e r e d a t low t e m p e r a t u r e s and

s h o r t a n n e a l i n g t i m e s , o r when n u c l e a t i o n p rob lems e x i s t o r i n t e r f a c e e n e r g i e s p l a y an

i m p o r t a n t r o l e . However, t h e s e p rob lems a r e m e n t i o n e d a t t h e a p p r o p r i a t e p l a c e s i n t he

t e x t where t h e r e a d e r i s r e f e r r e d to s p e c i a l i s e d l i t e r a t u r e on t h i s s u b j e c t .

- F i n i t e c o u p l e s i n which t h e s equence of t h e phase f o r m a t i o n as w e l l as the k i n e t i c s a r e

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

phases formed i n t h e b e g i n n i n g m i g h t be consumed d u r i n g f u r t h e r a n n e a l i n g . However,

g e n e r a l l y spoken t h e s e phenomena can be u n d e r s t o o d by a p p l y i n g t he t h e o r y f o r i n f i n i t e

d i f f u s i o n c o u p l e s u s i n g c o n t i n u o u s l y c h a n g i n g and members f o r such a c o u p l e .

- D i f f u s i o n i n i o n i c s y s t em s i n which an e x t r a c o n d i t i o n of e l e c t r i c n e u t r a l i t y a r i s e s , as

w e l l as d i f f u s i o n i n c o v a l e n t c e r s m i c s l i k e c e r t a i n c a r b i d e s and n i t r t d e s o r i n

c o m p l e t e l y o r d e r e d a l l o y s . In t h e s e c a s e s e s p e c i a l l y t h e d i f f i c u l t and o f t e n coup led

f o r m a t i o n and m i g r a t i o n of p o i n t d e f e c t s i s d i f f e r e n t from m e t a l l i c m a t e r i a l s . However,

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

m u l t l p h a s e d i f f u s i o n . The q u a l i t a t i v e i d e a s p r e s e n t e d i n t h i s p a p e r on t e r n a r y m u l t i p h a s e

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

2. MULTIPHASE DIFFUSION IN BINARY SYST~4S

Al though numerous s t u d i e s on m u l t l p h a s e b i n a r y d i f f u s i o n have bean p u b l i s h e d , s t i l l some

little known or even controversial issues exist. Generally, these are mostly mentioned only

as side remarks in papers in which another main subject is involved.

After a general introduction, a few typical examples of these issues are discussed in

this section in the following sequence:

- the time invariance of the concentration versus x/t ~ plot. The reasons for deviations are

discussed, as well as the use of deviating plots to get reliable diffusion data;

- the absence of bulk-equilibrium phases or the presence of non-equilibrlum phases in a

diffusion couple;

- deviations of interface concentrations measured in multlphase diffusion couples as

compared to b u l k - e q u i l i b r i u m v a l u e s i n homogenized a l l o y s ;

the p o s i t i o n s of t h e K i r k a n d a l l and Matano p l a n e s i n a d i f f u s i o n c o u p l e ;

the r o l e of t h e phase i n t e r f a c e ( s ) i n t he c r e a t i o n o r a n n i h i l a t i o n of v a c a n c i e s r e s u l t i n g

from t h e K i r k e n d a l l e f f e c t ;

t he morphology of t he phase i n t e r f a c e ( s ) i f g r a i n bounda ry d i f f u s i o n i s t a k i n g p l a c e ;

compar i son of t h e r e a c t i o n l a y e r t h i c k n e s s i n p l a n a r , c y l i n d r i c a l and s p h e r i c a l d i f f u s i o n

c o u p l e s ;

50 F . J . J . van Loo

- d i f f e r e n c e s i n morphology of r e a c t i o n zones i n t h e c e n t r e and a t t h e edges of a d i f f u s i o n

c o u p l e , i n c l u d i n g d i f f e r e n c e s d e p e n d i n g upon t he r e l a t i v e d i a m e t e r s of t he h a l f c o u p l e s |

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

- t h e e v a l u a t i o n of d i f f u s i o n c o e f f i c i e n t s i n sys t ems where l a r g e t o t a l volume changes

o c c u r ( e . g . d u r i n g i n t e r s t i t i a l d i f f u s i o n ) ;

- t he e v a l u a t i o n of d i f f u s i o n c o e f f i c i e n t s f o r v a p o u r - s o l i d c o u p l e s compared w i t h

s o l i d - s o l i d c o u p l e s .

Gene ra l i n t r o d u c t i o n i n t o m u l t i n h a e e b i n a r y diffusion

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

t he e v a l u a t i o n of d i f f u s i o n d a t a f rom c o n c e n t r a t i o n p l o t s i s t he same f o r s i n 6 1 e - p h a s e end

m u l t i p h a ~ e sy~ tams . The i n t e r d i f f u s i o n c o e f f i c i e n t D can be found from a c o n c e n t r a t i o n p l o t

by u s i n g t he N a t a n o - B o l t z m a n n a n a l y s i s a s g i v e n i n many e x c e l l e n t r e f e r e n c e books , such as

e . g . Adda and P h i l i b e r t [ 4 ] ~

We w i l l s t a r t t h i s i n t r o d u c t i o n by g i v i n g t h e n e c e s s a r y and u s e f u l e q u a t i o n s i n t h e i r

g e n e r a l form. At t h e end we p r e s e n t t h e s e e q u a t i o n s i n a s i m p l i f i e d form f o r c a s e s where

t h e s e can be u sed ( e . g . i f a s s u m p t i o n s of a c o n s t a n t t o t a l volume o r of e q u a l p a r t i a l molar

volumes a r e a l l o w e d ) .

The f i r s t d i f f i c u l t y i s t h e d e f i n i t i o n of t he d i m e n s i o n s f o r c o n c e n t r a t i o n and d i s t a n c e

in t h e s e p l o t s . In f a c t , t h e f r ame of r e f e r e n c e which i s chosen f o r t h e d e s c r i p t i o n of t he

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

indeed q u i t e a v a r i e t y of d e f i n i t i o n s e x i s t . As l ong as one i s work ing in a c o n s i s t e n t

System e v e r y e v a l u a t i o n i s e q u a l l y good and t he v a r i o u s d i f f u s i o n c o e f f i c i e n t s can be

t r a n s f o r m e d one i n t o a n o t h e r [ 5 ] . I t i s c o n v e n i e n t to d e f i n e t h e i n t e r d i f f u s i o n f l u x ~ f o r

component i by F i c k ' s f i r s t law i n t he form

~i " - D dcidx [m°les/m2s] ( I )

where D i s t h e i n t e r d i f f u s i o n c o e f f i c i e n t i n m 2 / s ; C i i s t h e c o n c e n t r a t i o n of component i i n

moles /m 3 and x i s t h e d i s t a n c e c o o r d i n a t e i n m.

In t h e c a s e o f a c o n s t a n t t o t a l volume t h e i n t e r d i f f u s i o n c o e f f i c i e n t D(C i ) a t a c e r t a i n

c o n c e n t r a t i o n Ci* can be found from t h e p e n e t r a t i o n p l o t of C i v e r s u s x by t h e

Matano-Bol tzmann e q u a t i o n :

D(C i ) = - ( ) ~ x d C i (2 ) C i

The meaning of t h e symbols f o l l o w s from F ig . 2 . 1 , f o r t h e i n t e r d i f f u s i o n of t he e l e m e n t s A

and B be tween two s t a r t i n g a l l o y s h a v i n g i n i t i a l B - c o n c e n t r a t i o n s of C B- and CB +,

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

Multiphase Diffusion 5 [

CB

C B

C B

i

, I I I

X ~ Xo, O

Fig . 2 .1 . C o n c e n t r a t i o n - p e n e t r a t i o n curve f o r component B in a d i f f u s i o n +

coup le be tween two a l l o y s w i t h s t a r t i n g c e m p o s i t i o n s CB and CB, r e s p e c t i v e l y .

The M a t a n o - r e f e r e n c e p l ane x o = 0 i s d e f i n e d in such a way that t he h o r i z o n t a l l y ha t ched

a r e a s a r e e q u a l . The f l u x d e f i n e d in t h i s way i s measured w i t h r e s p e c t to t h i s Matano p lane

o r , a l t e r n a t i v e l y , t o one of t he ends of t he coup le ( s i n c e the t o t a l volume i s c o n s t a n t ) .

The v a l u e of the i n t e g r a l i n Eq. (2) i s g iven by the d o u b l e - h a t c h e d a r e a in F ig . 2 .1 ; the

r e c i p r o c a l of the c o n c e n t r a t i o n g r a d i e n t has to be t aken a t t he p o s i t i o n x* where C i = Ci * .

If the total volume changes during the reaction (which means that the partial molar

volumes Vi a r e dependen t on c o n c e n t r a t i o n ) , Eq. (2) has to be r e p l a c e d by [ 6 , 7 ] :

x

where the c o n c e n t r a t i o n u n i t Y i s d e f i n e d as (N i - N i - ) / ( N i + - N i - ) ; N i i s t he mole f r a c t i o n

of component i , b e i n g N i - and Ni + a t the u n d i s t u r b e d l e f t - h a n d and r i g h t - h a n d ends of the

couple and V m i s the molar volume. In t h i s ca se t he f l u x i s measured w i t h r e s p e c t t o , say ,

the l e f t - h a n d end of t h e c o u p l e . A Matano p l ane x o t 0 canno t be u n i q u e l y d e f i n e d in t h i s

frame of r e f e r e n c e f o r a sys tem w i t h v a r y i n g t o t a l volume, s i n c e i t i s s p l i t up i n t o two

p l anes depend ing on whe the r t he l e f t - h a n d or the r i g h t - h a n d s i d e of t he coup le i s taken as a

r e f e r e n c e p o i n t . I t i s i ndeed n o t n e c e s s a r y to know a p l a n e x = 0 s i n c e in Eq. (3) t he

d i s t a n c e p a r a m e t e r a p p e a r s on ly in a d i f f e r e n t i a l form. Unless s t a t e d o t h e r w i s e , th roughout

t h i s paper a c o n s t a n t t o t a l volume ( c o n s t a n t b u t no t n e c e s s a r i l y equa l p a r t i a l molar

volumes) i s supposed in o r d e r to avo id n e e d l e s s c o m p l i c a t i o n s .

52 F..I.J. van Loo

The e q u a t i o n s ( 1 - 3 ) a r e r e l a t e d to i n t e r d i f f u s i o n phenmeena, c h a r a c t e r i z e d i n a b i n a r y

sys tem by t h e f a c t t h a t t h e same i n t e r d i f f u s i o n c o e f f i c i e n t D can be used f o r t he

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

d i f f u s i v i t i e s of b o t h components have to b e i n v e s t i g a t e d , t h e d i f f u s i o n f l u x e s r e l a t i v e to

some l a t t i c e p l a n e i n a phase have to be known. The o r i g i n of t h i s s o - c a l l e d K i r k e n d a l l

f rame of r e f e r e n c e can be found i n s i n g l e - p h a s e d i f f u s i o n c o u p l e s more o r l e s s e a s i l y by

using inert markers situated at the meeting interfaces at t m O. In a later paragraph the

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

The unequa l d i f f u s i o n f l u x e s o f t h e components w i t h r e s p e c t to t h e l a t t i c e p l a n e s i n a

phase i s compensa ted by a f l u x of v a c a n c i e s i n t h e s ane d i r e c t i o n as t h e f l u x o f t he s l o w e r

The r e l a t i o n be tween t h e i n t e r d i f f u s i o n f l u x ~ i i n t h e Natano and t h e i n t r i n s i c component.

diffusion flux Ji in the E/rkendall frames of reference is given by [8]:

~i = J i + Ci vKIM (4)

where v KIN e q u a l s t h e v e l o c i t y of K t r k e n d a l l f rame r e l a t i v e to t h e Matano f rame a t t h a t

p o s i t i o n and t ime where t h e c o n c e n t r a t i o n of component i e q u a l s C i .

Th i s v e l o c i t y v K/M can be d e f i n e d f o r e ach l a t t i c e p l a n e i n t h e d i f f u s i o n zone , bu t can on ly

be measured e a s i l y a t t h e l a t t i c e p l a n e c o r r e s p o n d i n g to t h e o r i s i u a l i n t e r f a c e a t t - 0 by

t he r e l a t i o n

v K/M = A x / Z t (5)

where 5x i s t h e m a r k e r d i s p l a c e m e n t , i . e . t h e d i s t a n c e be tween t h e K i r k e n d a l l p l a n e x K and

t he Matano p l a n e x o = 0.

The v e l o c i t y v K/M i s r e l a t e d to t h e f l u x o f v a c a n c i e s , wh ich can be s e e n from Eq. (4 ) and

t he r e l a t i o n s

~A~A + vs ~B = 0 (6)

VA JA + VB JB + Vm J v = 0 (6)

where t he s u b s c r i p t s A, B and V r e f e r t o t h e components A and B and to t h e v a c a n c i e s V. This

l e a d s to

v~/M = Vm Jv (7 )

S u b s t i t u t i o n i n t o gq. ( 4 ) , u s i n g t h e d e f i n i t i o n of t he mo la r volume

v m

m = C A + C B C A C B

leads t o :

~i = Ji + Ni Jv (8)

MuldphaseDiffusion

The i n t r i n s i c d i f f u s i o n c o e f f i c i e n t s D i a r e r e l a t e d to v K/M and t h e i n t e r d i f f u s i o u

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

v ~ I " = 0 a (D a - n s ) S Cal~x (9 )

53

= CA VA DB ÷ CB VB DA ( lO)

The v a l u e s f o r D i can be found a t t he c o n c e n t r a t i o n i n t he K i r k e n d a l l o r i g i n (marker

p o s i t i o n ) by combin ing Eqs. ( 2 , 5 , 9 ) and (10) w i t h e q u a t i o n [ 6 ] :

o~,- ~ ~,,~-o:_ c~ E°: ~ ~- - . i ,

I f t h e t o t a l volume does n o t r emain c o n s t a n t , gq. (11) has to be r e p l a c e d by [ 6 ] :

The r a t i o be tween t h e i n t r i n s i c d i f f u s i o n c o e f f i c i e n t s f o l l o w s from

Eq. (12) and i s e q u a l to

[ _

t_,,; In t h e r a t i o of

(laa)

("A-"P~ ÷ "; ~c.i--.^~ t h e i n t r i n s i c f l u x e s , which e q u a l s JA/JB = -DAVB/DBVA, o n l y t he

e x p r e s s i o n be tween b r a c k e t s r e m a i n s . Th i s e q u a t i o n c o n s i d e r a b l y s i m p l i f i e s i f NA+ = i and

N A- = 0.

The i n t r i n s i c d i f f u s i o n c o e f f i c i e n t s a r e r e l a t e d to t he t r a c e r d i f f u s i o n c o e f f i c i e n t s DiT

i n homogeneous a l l o y s o f t h e same c o m p o s i t i o n by [ 9 , 1 0 ] :

where d lnaA/d lnN A = d l n N B / d l n a B i s c a l l e d t he thermodynamic f a c t o r and a i i s t h e a c t i v i t y of

component i . The t e rm r A i s a c o r r e c t i o n f a c t o r o r i g i n a t i n g f r e e t h e v a c a n c y f l u x e f f e c t and

T i s d e p e n d e n t on t h e c r y s t a l s t r u c t u r e and t h e d i f f e r e n c e i n D i [ 1 1 ] . From Eq. [13] i t

f o l l o w s t h a t

D A D~ . 0 A . r A

T ~B D B D B • . r B ( I ~ )

and


T rB + NB D~ r A) d In a& = (NA DB " d i n N A

(15)

Since t he n u m b e r - f i x e d f rame of r e f e r e n c e , which i s d e f i n e d by a c o n s t a n t t o t a l amount of

moles a t e ach s i d e of t h e o r i g i n x - 0 , i s o f t e n used i n l i t e r a t u r e [12] i t i s c o n v e n i e n t to

g ive t he f o l l o w i n g r e l a t i o n s ( s u p e r s c r i p t N r e f e r s to t h e n u m b e r - f i x e d r e f e r e n c e f r a m e ) :

N T d I n a A D A = . d I n "

[161

For l a y e r s w i t h a v e r y s m a l l homogene i ty r e g i o n AN - 0 i t i s v e r y d i f f i c u l t to c a l c u l a t e

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

a p p r o a c h e s z e r o and t h e a p p a r e n t d i f f u s i o n c o e f f i c i e n t , t h e r e f o r e , becomes i n f i n i t e ( see

Eqs. (3) and ( 1 1 ) . A v e r y u s e f u l c o n c e p t f o r t h i s c a s e i s t he i n t e g r a t e d d i f f u s i o n

c o e f f i c i e n t d e f i n e d by Wagner [ 1 3 ] :

N 2 D int =.~ D d N = Day " A N (17)

where N 1 and N 2 a r e t h e - unknown - homogene i ty l i m i t s of t h e p h a s e , AN = N 2 - N 1 and Day i s

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

T h i s q u a n t i t y i s a m a t e r i a l c o n s t a n t l i k e t h e normal d i f f u s i o n c o e f f i c i e n t and i s s imply

r e l a t e d to t he p a r a b o l i c g rowth c o n s t a n t as d e f i n e d i n n e x t s e c t i o n .

The e q u a t i o n s g i v e n i n t h i s c h a p t e r can o f t e n be s i m p l i f i e d .

a ) In c a s e of a c o n s t a n t t o t a l vo lume, which means t h a t VA and VB a r e

c 0 n c e n t r a t i o n - i n d e p e n d e n t , Eqs. (2 ) and (11) can be used f o r f i n d i n g ~ o r D i . I f i n such a

coup le a t l e a s t one of t he s t a r t i n g m a t e r i a l s i s a pure component , e . g .

C i = 0, t h e n Eq. (11) r e d u c e s t o :

Di(xK)" + ~ (~Cl)x K & xK ¢idx (lla)

(Note t he s i m p l e i n t e g r a l which c o n t a i n s , c o n t r a r y t o Eq. ( 2 ) , t h e d i s t a n c e p a r a m e t e r as a

d i f f e r e n t i a l which means t h a t t h e p o s i t i o n of t h e Matauo p l a n e i s n o t i n v o l v e d [ 6 ] .


b ) In ca se of c o n s t a n t and e q u a l p a r t i a l mo la r vo lumes , which means VA = VB = Vm' s e v e r a l

e q u a t i o n s can be s l m p l l f l e d . In Eqs. ( 2 ) , ( I i ) end ( 1 1 a ) , C i can t h e n be r e p l a c e d by t h e mo!

f r a c t i o n N i . F u r t h e r :

vK/M = (D A - D B) ~NAI~¢ ( g a )

= NAD B + NBD A (lOa)

= ~N and D i = ~ (16a)

I f , on top of t h a t , NA = o and N:= 1 ( i . e . d i f f u s i o n be tween two p u r e m e t a l s ) , Eq. (12a) can

be s i m p l i f i e d to

DB " ~AdX / (l-NA)dX (12b)

~e time daoendence of the nenetratlon nlot

The a n a l y s i s g i v e n i n t he f o r e g o i n g p a r a g r a p h assumes t h a t t he same d i f f u s i o n p r o c e s s i s

r a t e - l i m l t l n g t h r o u g h o u t t he whole a n n e a l i n g p r o c e d u r e . In t h a t c a s e , a p l o t of t he

p e n e t r a t i o n c u r v e v e r s u s x / t ~ w i l l be i n v a r l a n t . The t r e a t m e n t i s s i m p l i f i e d by t a k i n g as a

d i f f u s i o n zone t he l a y e r of one compound growing be tween t he s t a r t i n g m a t e r i a l s , which

r e s t r i c t s t he problem to t h e s t u d y of t he l a y e r g rowth as a f u n c t i o n of t i m e . I f t he same

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

t h i c k n e s s d and t he a n n e a l l n g t ime t a r e r e l a t e d t h r o u g h t h e p a r a b o l i c growth c o n s t a n t kp as

d 2 = 2kp t . A p a r t from p o s s l b l e d i s t u r b i n g e f f e c t s d u r i n g t he h e a t i n g o r c o o l i n g p r o c e d u r e ,

s e v e r a l r e a s o n s may e x i s t which I n v a l i d a t e t h i s a s s u m p t i o n . A few of t h e s e a r e d i s c u s s e d

h e r e .

a ) The m a t i n g s u r f a c e s may i n i t i a l l y be cove red by a t h i n l a y e r of d i f f e r e n t c o m p o s i t i o n ,

e . g . an ox ide f i l m o r a s e g r e g a t i o n l a y e r o r i g i n a t i n g from i m p u r i t i e s p r e s e n t i n t he

starting'materials. If an oxide film at the interface acts as a barrier for diffusion, then

some incubation time will pass before an undisturbed diffusion process can begin. The plots

of the width of the diffusion layer d and d 2, respectlvely, versus time then resemble those

shown in Fig. 2.2.

d

d 2

T

5 I0 15 Pt

(a)

Fig . 2 . 2 .

tOG

200

0

(b)

5 10 15


P l o t s of l a y e r t h i c k n e s s d ( a ) and d 2 (b ) v e r s u s t ime i n a r b i t r a r y u n i t s i n a m u l t i p h a s e d i f f u s i o n c o u p l e , i f an i n c u b a t i o n t ime i s i n v o l v e d b e f o r e t he r e g u l a r d i f f u s i o n p r o c e s s s e t s i n .

An example of t h i s phenomenon was found i n t he Ti-A1 sys tem [1~] . An o x i d e l a y e r be tween

Ti and A1 b r o k e down a f t e r a p e r i o d of many h o u r s , d u r i n g which a s low l i n e a r

r e a c t i o n - l i m i t e d g rowth of TiA13 was found . A f t e r t h i s i n c u b a t i o n t ime t h e l a y e r grew v e r y

f a s t and p a r a b o l i c a l l y w i t h t ime i n a d i f f u s i o n - l i m i t e d p r o c e s s . A row of p o r e s i n t h e l a y e r

marked t h e t r a n s i t i o n p o i n t i n t h e p r o c e s s , i n d i c a t i n g t he r e s i d u e of t h e d i s t u r b i n g o x i d e

film.

A s e g r e g a t i o n l a y e r a t t h e i n t e r f a c e i s o f t e n p r e s e n t as a r e s u l t of h e a t i n g t he coup le

o r p r e a n u e a l i n g t h e s t a r t i n g m a t e r i a l s . E s p e c i a l l y a toms such as c a r b o n o r p h o s p h o r u s ,

p r e s e n t i n o n l y a few p a r t s p e r m i l l i o n i n t h e b u l k m a t e r i a l , may g i v e r i s e to e n r i c h e d

s u r f a c e l a y e r s of o n l y a few n a n o m e t e r s which i n f a c t change t he b i n a r y sys tem i n t o a

t e r n a r y one . The d i f f u s i o n p r o c e s s can t h e n be t o t a l l y d i f f e r e n t , b u t a l s o t i m e - d e p e n d e n t

b e c a u s e of t he l i m i t e d amount of t he s e g r e g a t i n g e l e m e n t [ 1 5 , 1 6 ] .

b ) The r e a c t i o n be tween e l e m e n t s to form a new compound may be t h e r a t e - l l m l t l n g s t e p . The

l a y e r t h e n s t a r t s g rowing l i n e a r l y w i t h t i m e , b u t a f t e r r e a c h i n g a c e r t a i n t h i c k n e s s t he

d i f f u s i o n p r o c e s s e v e n t u a l l y becomes r a t e - l i m i t i n g . Th i s c a s e i8 e x t e n s i v e l y d i s c u s s e d by

Dybkov [ 1 ] . The p l o t s of d and d 2, r e s p e c t i v e l y , v e r s u s t ime w i l l t h e n be l i k e shown i n

F ig . 2 . 3 .

In f a c t , i n b u l k d i f f u s i o n c o u p l e s no c l e a r l y p r o v e n c a s e of r e a c t i o n - l i m i t e d g r o v t h i s

known to t h e a u t h o r f o r n o n - p o r o u s r e a c t i o n l a y e r s . However, i t must i m m e d i a t e l y be added

t h a t t he d 2 / t p l o t s i n most s t u d i e s on n m l t i p h a s e d i f f u s i o n show i n - e n o u g h measured p o i n t s

and too l a r g e a s p r e a d to be s u r e of t he e x a c t r e l a t i o n be tween t he l a y e r t h i c k n e s s and

t ime . Small i n c u b a t i o n t i m e s o r p e r i o d s of n o n - p a r a b o l l c g rowth a r e t h e n e a s i l y o v e r l o o k e d .

15

d 2

t 20(

d

T 10

/

J 0 5 10 15

P t (a)

100

I I t

5 10 15 -----~ t

(b)

Multiphas¢ Diffusion 57

Flg. 2.3. P l o t s o f l a y e r t h i c k n e s s d ( a ) and d 2 (b ) v e r s u s t ime i n a r b t t r a t y u n i t s i n a m u l t i p h a s e d i f f u s i o n c o u p l e , i f t h e r e a c t i o n of compound f o r m a t i o n i s the r a t e - l i m i t i n g s t e p i n t h e b e g i n n i n g of t h e p r o c e s s .

In porous l a y e r s , t h e r e a c t l o n - l i ~ i t e d l l n e a r g rowth i s w e l l known. In e x p e r i m e n t s In

which t he s u b s t r a t e r e a c t s w i t h a s u r r o u n d i n g gas o r l l q u l d , r e a c t l o n - l ~ i t e d and

d l f f u s l o n - l l s l t e d g rowth can be d l s t l n g u l s h e d by t h e morphology change o f an o r l g l n a l l y

c u b i c s u b s t r a t e a s shown i n F ig . 2 . 4 , t a k e n from [ 1 7 ] . In t he f i r s t c a s e , t he t y p l c a l

' ~ a l t e s e c r o s s " t ype f i g u r e forms from t h e s h a r p c o r n e r s of t h e s u b s t r a t e , s i n c e t h e gas o r

l i q u i d can r e a c t w i t h t h e s u b s t r a t e i n d e p e n d e n t o f t h e t h i c k n e s s of t h e porous r e a c t i o n

l a y e r a l r e a d y fo rmed . In t h e l a t t e r c a s e , volume d i f f u s i o n of t h e s u b s t r a t e atoms t h r o u g h

t he r e a c t i o n l a y e r towards t h e gas phase i s r a t e - l i m i t i n g , l e a d i n g to comple t e cove rage and

rounded e d g e s . Exsmples o f t h e c r o s s - t y p e a t t a c k a r e t h e o x i d a t i o n of m e t a l s such as Nb o r

Ta and of Co-~/C a l l o y s [ 1 7 ] , t he h o t - d i p a l o m i n t z i n g o f t i t a n i u m [18] and t h e h o t - d i p

g a l v a n i z i n g of i r o n [ 1 9 ] .

Flg . 2.4 .

JP~SC 20:I-E

r : ~ ¸ 1

i

~ .,~.'." (a)

(b)

M o r p h o l o g i e s o c c u r r i n g d u r i n g t h e o x i d a t i o n of an i n l t l a l l y c u b i c s u b s t r a t e in c a s e o f l l n e a r ( a ) and p a r a b o l i c g rowth (b ) [ 1 7 ] .

58 F.J.J. van Loo

c) In many d i f f u s i o n couple expe r imen t s the l a y e r growth i s f a s t e r in the b e g i n n i n g of the

p roces s because of t he sma l l g r a i n s p r e s e n t a t the s t a r t which l ead to a f a s t g ra in -boundary

d i f f u s i o n p r o c e s s . During the a n n e a l i n g p rocedure , g r a i n growth o c c u r s , and e v e n t u a l l y the

s lower volume d i f f u s i o n p roces s w i l l be the r a t e - l i m i t i n g s t e p . P l o t s of d and d 2 ve r sus

t ime then appear as shown in F i g . 2 .5 .

t,00

d s" ~ j, j l ~ / I 200

/ / t / f

/ / / // / / / 1 0 0 ~

! / ! /

a l J , t ~ / / I n S 0 t. t 5 10 5 0 5 P t 110

a) 0~)

Fig . 2 .5 . P l o t s of l a y e r t h i c k n e s s d (a ) and d 2 (b) v e r s u s t ime in a r b i t r a r y u n i t s in a multlphaee diffusion couple, if in the beginning of the diffusion process fast g r a i n - b o u n d a r y d i f f u s i o n o c c u r s .

I t i s i n t e r e s t i n g to i n q u i r e whether a d 2 / t o r a d / t ~ p l o t can b e t t e r be used in o rde r to

d e s c r i b e the p rope r v a l u e s f o r the s t e a d y - s t a t e p a r a b o l i c r a t e c o n s t a n t kp in the second

growth s t a g e . P i e r a g g i [20] conc luded in an i n t e r e s t i n g d i s c u s s i o n t h a t a d / t ~ p l o t has to

be p r e f e r r e d i f t he l a y e r , formed dur ing an i n i t i a l t r a n s i e n t p e r i o d o f f a s t e r k i n e t i c s ,

does no t c o n t r i b u t e to the s t e a d y - s t a t e r a t e c o n t r o l i n the l a t e r p a r a b o l i c s t a g e . This

s i t u a t i o n o f t e n a r i s e s in o x i d a t i o n r e a c t i o n s . In t h a t c a s e , the r e l a t i o n

dp 2 - (d - d l ) 2 - 2 kp ( t - t l ) (18)

a p p l i e s , in which d and t a r e the t o t a l t h i c k n e s s and t ime , r e s p e c t i v e l y , and d i i s the

thickness after the transient period ending at time t i after which the steady-state

parabolic growth starts. The value dp is the thickness of the part of the layer, grown in

the p a r a b o l i c s t a g e a f t e r t ime t i . This r e l a t i o n l e ads t o :

d = d i + {2 kp ( t - t l ) } ~ (18a)

A p l o t of d and dp v e r s u s t ~ f o r a s p e c i f i c case i s g iven in F i g . 2 .6a . I t shows n e a r l y

p a r a l l e l p l o t s f o r d and dp a f t e r l o n g e r a n n e a l i n g t i m e s , l e a d i n g to a n e a r l y c o r r e c t v a l u e

f o r kp from the s l o p e of an e x p e r i m e n t a l d / t ~ p l o t . This i s t r u e f o r cases where t i i s no t

too l a r g e compared to t ; the appa ren t v a l u e kp ' from the d / t ~ p l o t and the r e a l v a l u e kp a r e

r e l a t e d through [20]

Muldphase Diffusion 59

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Fig . 2 . 6 . P l o t s of d v s . t ~ ( a ) and d 2 v s . t ( b ) i n a r b i t r a r y u n i t s i n c a s e of t he p r e s e n c e of an i n i t i a l l a y e r of t h i c k n e s s d = 2 a f t e r a t r e n s i a n t p e r i o d t = 10, which does n o t c o n t r i b u t e to t h e r a t e - c o n t r o l i n t he f o l l o w i n g p a r a b o l i c s t a g e w i t h

= 0 . 0 5 . A v a l u e k_ ' would have b e e n found from t h e " e x p e r t m e n t a l " c u r v e s . ~ ( c ) and (d ) t h e J IBe p l o t s a r e shown, now f o r t he c a s e i n which t h e i n i t i a l l a y e r does c o n t r i b u t e to t he f o l l o w i n g p a r a b o l i c g rowth s t a g e .

kp ' = k p t / ( t - t i ) (19)

which i s i n d e p e n d e n t of t h e v a l u e of d i .

I f f o r t h i s c a s e a d 2 / t p l o t had bean used , a wrong v a l u e f o r s t e a d y - s t a t e kp would have

bean found as can be s e e n by t h e d i f f e r e n t s l o p e s of d 2 / t and d 2 p / t i n F i g . 2 . 6 b . The

a p p a r e n t v a l u e kp ' f rom t he d 2 / t p l o t i s t h e n r e l a t e d to t h e t e a l v a l u e of kp t h r o u g h :

kp ' - kp [ I + d l / { 2 kp ( t - t i ) } ~ ] (20)

as shown by P t e r a g g t [ 2 0 ] . In F ig . 2 .6 t he s h a r p t r a n s i t i o n p o i n t be tween t he two growth

reg imes i s smoothed o u t to a c o n t i n u o u s c u r v e .

On t he o t h e r hand , i f t h e v a l u e of t he t h i c k n e s s d i a f t e r t he t r a n s i e n t p e r i o d does

c o n t r i b u t e to t he s t e a d y - s t a t e p a r a b o l i c growth i n t h e second s t a g e , t h e n t he f o l l o w i n g

r e l a t i o n a p p l i e s :

d 2 - d i 2 ~ 2 kp ( t - t i ) (21)


In t h i s c a s e , a d 2 / t p l o t l e a d s to t h e c o r r e c t v a l u e of kp , whereas a d / t ~ p l o t l e a d s Co

e r r o n e o u s r e s u l t s as shown i n F i g . 2 .6 c and d w i t h an a p p a r e n t kp ' a c c o r d i n g to

kp ' = kp t / ( t - t i + d i 2 / 2 k p ) (22)

The same i s t r u e i n c a s e s where an i n c u b a t i o n t ime i s i n v o l v e d w i t h o u t a s u b s t a n t i a l

t r a n s i e n t l a y e r t h i c k n e s s l i k e i n F i g . 2 . 2 . In t h a t c a s e , d i i s z e r o and a d 2 / t p l o t w i l l

g ive t he p r o p e r k p - v a l u e , whe reas a d / t ~ p l o t l e a d s to an a p p a r e n t v a l u e kp ' g i v e n by

Eq. ( 1 9 ) . I t i s , t h e r e f o r e , d e p e n d e n t on t h e k i n e t i c r o l e of t h e t r e n s i e n t l a y e r and t he

r e l a t i v e t h i c k n e s s e s d and d i w h e t h e r a d 2 / t o r a d / t ~ p l o t i s p r e f e r r e d when k i n e t i c d a t a

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

d i f f i c u l t to use b e c a u s e of t h e p o s s i b l e l a r g e s p r e a d i n l a y e r t h i c k n e s s measu remen t s . I t i s

t hen n e c e s s a r y to i n v e s t i g a t e t h o r o u g h l y m o r p h o l o g i c i t e m s l i k e p o r o s i t y and g r a i n s i z e i n

o r d e r to r e d u c e t he i n f l u e n c e o f p r e j u d i c e i n t h e e v a l u a t i o n o f t h e e x p e r i m e n t a l r e s u l t s !

Phases formed i n d i f f u s i o n e o u v l e s compared w i t h t h e p h l s a d lae r~m

Th i s c o n s i d e r a t i o n i s r a t h e r c l o s e l y r e l a t e d to t h e f o r m e r one , s i n c e e s p e c i a l l y i n

l i t e r a t u r e on t h i n - f i l m r e a c t i o n d i f f u s i o n a s e q u e n t i a l n u c l e a t i o n of v a r i o u s p h a s e s has

been r e p o r t e d [ 2 1 ] . T h i s , of c o u r s e , r u l e s o u t p a r a b o l i c growth of a l l l a y e r s s t a r t i n g a t

t = 0. S e v e r a l c a s e s w i l l be d i s c u s s e d :

a ) An o b v i o u s r e a s o n f o r t h e a b s e n c e o f an e q u i l i b r i u m phase i s t h e p r e s e n c e of b a r r i e r

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

s t a r t i n g m a t e r i a l s . In t h e l a t t e r c a s e , s e g r e g a t i o n o f i m p u r i t i e s , p r e s e n t maybe o n l y i n the

ppm-renge i n one of t h e end members , can c a u s e an e n r i c h m e n t i n t h e d i f f u s i o n zone . Th i s

g i v e s r i s e to a t e r n a r y s y s t e m , and t h e n t h e d i f f u s i o n p a t h migh t e a s i l y miss a b i n a r y

e q u i l i b r i u m p h a s e , as i s shown l a t e r .

On t h e o t h e r h a n d , t h e same e f f e c t can c a u s e t h e f o r m a t i o n of a n o n - e q u i l i b r i u m p h a s e ,

s t a b i l i z e d by i m p u r i t i e s . N o t o r i o u s examples a r e c a r b i d e s l i k e Mosgi3C [22] o r Mo6Ni6C [15] ,

which migh t be c o n f u s e d w i t h t h e p u r e l y b i n a r y p h a s e s MosSi 3 and MoNi. The same i s t r u e f o r

o x i d e s l i k e Ti4Ni20 and Ti~Fe20, which m i g h t b e c o n f u s e d w i t h t h e b i n a r y compound Ti2Ni o r

w i t h a n , i n f a c t n o n - e x i s t i n g , b i n a r y "compound" Ti2Fe [ 2 3 ] . Th i s i s e s p e c i a l l y c o n f u s i n g

when t h e s e t e r n a r y p h a s e s grow i n t h i n l a y e r s i n s e r i e s w i t h g e n u i n e b i n a r y t h i c k e r l a y e r s .

An e a s y way to v e r i f y t h e p u r e l y b i n a r y c h a r a c t e r of an e q u i l i b r i u m l a y e r i s t he use of

i n c r e m e n t a l c o u p l e s . The c o m p o s i t i o n s of t h e end members a r e t h e n chosen q u i t e c l o s e to t he

a p p a r e n t phase i n q u e s t i o n , i n such a way t h a t o n l y t h i s phase m i g h t be formed. The end

members may be s i n g l e - p h a s e f o r two-phase a l l o y s . A t r u e e q u i l i b r i u m phase t h e n grows

p a r a b o l i c a l l y w i t h t ime as a r e l a t i v e l y t h i c k l a y e r , whereas an i m p u r i t y - s t a b i l i z e d phase

s t o p s growing a f t e r some t ime s i n c e t he i m p u r i t y i s t o t a l l y w i thd rawn from t h e end members.

F ig . 2 .7 s c h e m a t i c a l l y i l l u s t r a t e s t h i s p r o c e d u r e .


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Fig . 2 . 7 . In ( a * d) a r e shown t he phase d iag ram of an A-B s y s t e m , t he p e n e t r a t i o n p l o t s f o r component B i n a c o u p l e A/B and y /B , and t h e p l o t of dg 2 v e r s u s t ime i n t he y/B c o u p l e . In ( a ' * d ' ) t h e g - p h a s e i s a n o n - e q u i l i b r i u m , i m p u r i t y - s t a b i l i z e d compound. The growth of t he g - l a y e r i n a coup le y/B t h e n r e a c h e s soon a maximum v a l u e .

b ) E s p e c i a l l y i f t h i n f i l m s a r e fo rmed , e i t h e r a t t h e b e g i n n i n g o f t h e e x p e r i m e n t o r because

of a v e r y low d i f f u s i v i t y i n the p h a s e , t he f o r m a t i o n of l a y e r s i n t h e d i f f u s i o n zone migh t

be governed by t h e n u c l e a t i o n of phases [ 1 , 2 1 , 2 4 , 2 5 ] . D ' H e u r l e [21] g i v e s a s u r v e y on t h i s

s u b j e c t , t r e a t i n g a number of s i l i c t d e sys t ems where n u c l e a t i o n may be sO d i f f i c u l t t h a t i t

domina te s t he p r o c e s s of phase f o r m a t i o n . Th i s s i t u a t i o n may a r i s e i f t he d r i v i n g f o r c e

(change i n f r e e e n e r g y ~G) to b u i l d a new phase i s v e r y low and migh t become ze ro by t he

c o u n t e r a c t i n g i n f l u e n c e of the i n c r e a s e i n s u r f a c e e n e r g y . The phase then does n o t form a t

a l l ; a t h r e s h o l d v a l u e of ~G = -~00 J /cm 3 i s i n d i c a t e d [21 ] . At a s p e c i f i c h i g h e r

t e m p e r a t u r e , t h e f o r m a t i o n of t h e new compound i s f a v o u r e d and growth t hen s t a r t s

p a r a b o l i c a l l y w i t h t ime .

c ) I f l o c a l e q u i l i b r i u m i s e s t a b l i s h e d i n t h e d l f f t m l o n zone , e ach l a y e r has to show up from

the b e g i n n i n g of t he e x p e r i m e n t . Wagner [13] and Gurov e t a l . [26] d e r i v e d e q u a t i o n s f o r the

growth of each phase i n a b i n a r y c o u p l e , s t a r t i n g from the pu re components A end B, of t he

type

N~ d 2 d

f "A = K1 • ÷ K2 • - 2t (23)


in which D(y) i s the i n t e r d i f f u s i o n c o e f f i c i e n t in the phase y between the boundary

c o n c e n t r a t i o n s N A' and NA" and d¥ i s the l a y e r t h i c k n e s s of phase y. K 1 and K 2 a r e f u n c t i o n s

of the c o n c e n t r a t i o n s and d i f f u s i o n c o e f f i c i e n t s in the o t h e r phases . S ince the l e f t - h a n d

p a r t of Eq. (23) i s always p o s i t i v e , each phase has to grow to some ( p o s s i b l y v e r y sma l l )

t h i c k n e s s d 7. Kidson [2] p u b l i s h e d an o f t e n quoted paper which r eaches the same c o n c l u s i o n ;

p u r e l y from an e q u i l i b r i u m d i f f u s i o n - l i m i t e d growth concept a l l phases have to be p r e s e n t ,

a l b e i t in v e r y sma l l wid ths ( see n e x t i t e m ) . The p u r e l y a l g e b r a i c n e E a t i v e growth c o n s t a n t s

f o r a phase ment ioned by Kidson may no t be i n t e r p r e t e d as a reason f o r the n o n - e x i s t e n c e of

a l a y e r of t h a t phase .

I f , however , a r e a c t i o n - l i m i t e d growth o f some phase i s i n v o l v e d , i t s fo rma t ion can be

suppressed a c c o r d i n g to Dybkov I l l , l e a d i n g to an i n c u b a t i o n p e r i o d f o r the growth of t h i s

phase . The r e a d e r has to be aware t h a t t h i s has n o t h i n g to do w i t h n u c l e a t i o n problems

p r e v i o u s l y ment ioned , bu t i t r e s u l t s from p u r e l y k i n e t i c r easons as proposed by Dybkov. In

the a u t h o r ' s o p i n i o n , t h i s p r e d i c t i o n has y e t to be proven e x p e r i m e n t a l l y and w i l l be one of

the c h a l l e n g e s f o r a c c u r a t e e x p e r i m e n t a l i s t s , e s p e c i a l l y w i t h r ega rd to the n e x t i tem.

d) Sometimes phases seam to be m i s s i n g in d i f f u s i o n coup le s when i n v e s t i g a t e d by mic roscop ic

o r microprobe a n a l y s i s . In t h a t ca se they might a c t u a l l y be p r e s e n t in v e r y t h i n l a y e r s ,

o b s e r v a b l e o n l y by e l e c t r o n mic roscopy . Examples which need a thorough a n a l y s i s in t h i s

r e s p e c t a r e t he d i f f u s i o n coup l e s Ti-A1 [14] and Cu-Si [16] . In bo th sys tems v a r i o u s phases

should be p r e s e n t a c c o r d i n g to the phase d iagram, bu t on ly one phase seems to be p r e s e n t in

the d i f f u s i o n c o u p l e s , v i z . TiA13 and Cu3Si. However, c a l c u l a t i o n s show t h a t the miss ing

phases under the e x p e r i m e n t a l c o n d i t i o n s should be p r e s e n t in t h i c k n e s s e s f a r below 1

mic romete r and a r e , t h e r e f o r e , m i c r o s c o p i c a l l y no t o b s e r v a b l e . The reason i s , t h a t the

d i f f u s i v i t y in the ' ~ i s s i n $ " phases i s s e v e r a l o r d e r s of maEnitude lower than those in TiA13

and Cu~Si.

In the t i t a n i u m - a l u m i n i o m c a s e , the m i s s i n g phases (TiA12, TiA1 and Ti3A1) were grown in

an i n c r e m e n t a l couple T i /T iA13 . A f t e r t h e i r growth to r e l a t i v e l y t h i c k l a y e r s the remaining

couple was p l aced v e r s u s A1 and annea led a g a i n . As e x p e c t e d , the l a y e r s TiA12, TiA1 and

Ti3A1 m i c r o s c o p i c a l l y d i s a p p e a r e d and on ly TiA13 cou ld be observed [14] . Whether these

l a y e r s remained p r e s e n t in v e r y t h i n l a y e r s i s now b e i n g i n v e s t i g a t e d in our l a b o r a t o r y by

means of e l e c t o n mic roscopy .

I n t e r f a c e concentrations in d i f f u s i o n couo l e s comoared w i t h b u l k - e o u i l i h r i u m v a l u e s

S ince the q u a n t i t a t i v e e v a l u a t i o n of d i f f u s i o n coup le s i s o f t e n used f o r phase diagram

d e t e r m i n a t i o n , i t i s impor t an t to be su re t h a t b u l k - e q u i l i b r i u m v a l u e s a r e r e a l l y a t t a i n e d

a t the i n t e r f a c e s .

I f the d i f f u s i o n coup le t e c h n i q u e i s no t used in a p roper way, a number of e r r o r sources

might appea r . Some of them were a l r e a d y ment ioned b e f o r e , such a s :

- the l a r g e r o l e p layed by i m p u r i t i e s

- the n e c e s s i t y to use i n c r e m e n t a l coup le s because o f the problem of m i s s ing phases .

Apar t from t h e s e , some p r a c t i c a l d i f f i c u l t i e s may a r i s e , such a s :

a] S teep c o n c e n t r a t i o n g r a d i e n t s may occur in growing phases o r in the end members of the

coup l e . By microprobe a n a l y s i s i t i s d i f f i c u l t to measure r e l i a b l e c o n c e n t r a t i o n v a l u e s a t a

d i s t a n c e s h o r t e r than about 2 pm from the i n t e r f a c e . This means, t h a t an e x t r a p o l a t i o n i s

always n e c e s s a r y which might cause e r r o r s when s t e e p g r a d i e n t s a r e i n v o l v e d . Again,

i n c r e m e n t a l coup le s have to be used o r an a n a l y s i s of e q u i l i b r a t e d two-phase a l l o y s has to

be made in o r d e r to g e t r e l i a b l e r e s u l t s .

Multiphase ~iffusion 63"

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

a f f e c t t he c o n c e n t r a t i o n measu remen t s a t d i s t a n c e s as l a r g e as 40 ~m from t h e i n t e r f a c e .

C o r r e c t i o n s f o r t h i s e f f e c t a r e p o s s i b l e b u t d i f f i c u l t [ 2 7 ] . In a b i n a r y c o u p l e , u s u a l l y one

of the e l e m e n t s does n o t s u f f e r f rom t h e f l u o r e s c e n s e e f f e c t and one cou ld measure the

c o n c e n t r a t i o n of t h a t e l e m e n t by m i c r o p r o b e a n a l y s i s to d e s c r i b e t he p e n e t r a t i o n p l o t . I f ,

f o r i n s t a n c e , pu re Cu and Co a r e clamped t o g e t h e r w i t h o u t any d i f f u s i o n t a k i n g p l a c e , Co-K~

r a d i a t i o n can a p p a r e n t l y be measured up to 40 pm in t he pure Cu. About ~ .8 mole • Co appea r s

to be in Cu at the interface [27]! Measuring Co-K= radiation diminishes the error, but still

an apparent concentration of 2 mole % Cu is found in Co at the interface, whereas the

Cu-radlation can still be detected at 15 ~n from the interface in Co.

c] As shown f o r d i f f u s i o n c o u p l e s i n t h e Cu-A1 sys t em [ 2 8 ] , s low c o o l i n g r a t e s may e a s i l y

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

d] Adds and P h i l i b e r t [4] g i v e a number of examples i n which t h e use of an e x t e r n a l p r e s s u r e

changes t he k i n e t i c s of t h e l a y e r growth as w e l l as t he i n t e r f a c e c o n c e n t r a t i o n s . Large

p r e s s u r e s l e a d to c o n c e n t r a t i o n v a l u e s which a r e i d e n t i c a l w i t h e q u i l i b r i u m b u l k phases

g i v e n by t he phase d i ag ram. Acco rd ing to the a u t h o r ' s e x p e r i e n c e , an e x t e r n a l p r e s s u r e of

t y p i c a l l y 30 kg/cm 2 ( d e p e n d i n g on t h e a n n e a l i n g t e m p e r a t u r e ) i s o f t e n needed to g e t r e l i a b l e

and r e p r o d u c i b l e r e s u l t s . The e x t e r n a l p r e s s u r e i s t h o u g h t to e s t a b l i s h good c o n t a c t between

t he end members ( 1 6 , 2 9 , 3 0 ] .

From a t h e o r e t i c a l p o i n t of v iew a more i n t e r e s t i n g q u e s t i o n i s w h e t h e r a s u p e r s a t u r a t i o n

of one of t h e e l e m e n t s must o c c u r b e c a u s e of t he need f o r a d r i v i n g f o r c e f o r t he l a t t i c e

t r a n s f o r m a t i o n [ 3 ] . A l so a s l i g h t s u p e r s a t u r a t i o n of v a c a n c i e s s h o u l d be n e c e s s a r y to

m a i n t a i n a vacancy f l u x [ 3 1 ] . A l t h o u g h v e r y s l i g h t d i f f e r e n c e s from b i n a r y e q u i l i b r i u m

v a l u e s a r e p r e d i c t e d , in ou r l a b o r a t o r y we have n e v e r o b s e r v e d d i f f e r e n c e s which exceeded

the e x p e r i m e n t a l i n a c c u r a c y . Remig [32] came to t he same c o n c l u s i o n i n a s u r v e y pape r on

t h i s s u b j e c t . An o f t e n quo ted e x p e r i m e n t a l p r o o f f o r l a r g e d i s c r e p a n c i e s be tween

c o n c e n t r a t i o n s measured i n d i f f u s i o n c o u p l e s and i n e q u i l i b r a t e d a l l o y s was p u b l i s h e d by

E i f e r t e t a l . [ 3 ] f o r t h e Cu-A1 s y s t e m . Newly r e p o r t e d r e s u l t s [28] showed t h a t the

d i s c r e p a n c i e s found by E i f e r t e t a l . p r o b a b l y o r i g i n a t e d from the s low c o o l i n g r a t e s in

t h e i r e x p e r i m e n t s .

The o o s i t i o n of t h e K l r k e n d a l l and Natano v l a n e s i n a d l f f t m l o n c o u n l e

Let us c o n s i d e r t he f o r m a t i o n of Ni3Sn 2 be tween t h e a d j a c e n t p h a s e s Ni3Sn 4 and Ni3Sn ,

which has been s t u d i e d by van Beek e t a 1 . [ 3 3 ] . A f t e r a n n e a l i n g a t 792"C the r e s u l t i s

s c h e m a t i c a l l y shown i n F ig . 2 . 8 .

The Matano p l a n e x o can be found g r a p h i c a l l y from the p e n e t r a t i o n cu rve as t r e a t e d

b e f o r e , o r e x p e r i m e n t a l l y e . g , by p r e v e n t i n g t h e r e a c t i o n be tween Ni3Sn 4 and NiHSn a t the

edges of t h e c o u p l e [34] as shown i n F i g . 2 .8 o r by o t h e r i n g e n e o u s methods [35] .

For s i m p l i c i t y , l e t us c o n s i d e r t he p h a s e s i n t he c o u p l e as p u r e l y s t o i c h i o m e t r i c

l i n e - c o m p o u n d s . Then, t he p o s i t i o n of t he Natano p l a n e i n t h e N i3Sn2-1aye r can immed ia t e ly

be found by t he r e a c t i o n e q u a t i o n

F. J. J. van Loo

Ni3Sn 4

I--a-- Ni3Sn2 . . . . . X O

'=---- b XK

x 2 d 2

Ni3Sn

Fig . 2 .8 . Schemat ic drawing of a Ni3Sn&/Ni3Sn d i f f u s i o n coup le in which the phase N i ~ a 2 i s formed [33] .

volume r a t i o

2 Ni3Sn + Ni3Sn ~ • 3 Ni3Sn 2

103 : 125 : 225

T h e r e f o r e , d2/d 1 = 1251103 = 1 .21 . Because h a r d l y any change in t o t a l volume o c c u r s , the

p a r t i a l molar volume8 VNi and VSn a r e n e a r l y c o n s t a n t , v i = . about 9 and 24.

The K i r k e n d a l l marke r s , put i n t he o r i g i n a l i n t e r f a c e a t t = 0, were d i s p l a c e d towards

the Ni3Sn-s ide a t the p o s i t i o n xK, l e a d i n g to a r a t i o x 2 / x 1 = 1 .85 .

From Table 2 .1 i t i s c l e a r t h a t t h i s r a t i o e q u a l s 0 .5 on ly i f Sn d i f f u s e s and Ni i s

immobile ( l i n e a in F i g . 2 .8) s i n c e then a t t he N i 3 S n - s i d e , two t imes as many Ni3Sn 2

molecu les a r e formed. I f o n l y Ni d i f f u s e s and Sn i s immobile t h i s r a t i o x 2 / x 1 = 2 ( l i n e b ) .

From the se d a t a i t f o l l o w s t h a t Ni d i f f u s e s f a s t e r than Sn.

Table 2.1

Only Sn d i f f u s e s Only Ni d i f f u s e s

Nt3Sn4-s ide Ni3Sn 4 • Ni3Sn 2 ÷ 2 8n Ni3Sn 4 ÷ 3 Ni ~ 2 Ni3Sn 2

Ni3Sn-s lde 2 Ni3Sn ÷ 2 Sn "*' 2 Ni3Sn 2 2 Ni3Sn ~ Ni3Sn 2 + 3 Ni

x21x I 0 .5 2

Multiphase Diffusion 6S

I f b o t h Ni and Sn d i f f u s e , t h e n the s i m p l e c h e m i c a l e q u a t i o n s of Tab le 2 .1 a r e dangerous

to uJe i f t h e p a r t i a l m o l a r volumes a r e n o t e q u a l . In f a c t , t h e m a t h e m a t i c a l l y c o r r e c t

Eq. (12a) a c c o u n t s f o r t h i s d i f f e r e n c e i n a tomic volume as i t does f o r t h e p r e s e n c e of a

homogene i ty r a n g e of a b o u t 5 mole ~ i n Ni3Sn 2 end t h e n o n - s t o i c h i o m e t r y o f t he end members.

The e x p e r i m e n t a l l y d e t e r m i n e d v a l u e of x 2 / x 1 = 1.85 t h e n l e a d s to a r a t i o DNi/Dsn of abou t

1 .5 . I f o n l y t h e d i f f e r e n c e i n a t o m i c volume had been t a k e n i n t o a c c o u n t , t he r a t i o DNi/Dsn

would have b e e n c a l c u l a t e d as 9 . 5 .

Th i s a n a l y s i s has been g i v e n i n some d e t a i l to show:

a] The r a t i o d 2 / d l , i . e . t h e r e l a t i v e p e n e t r a t i o n of t h e r e a c t i o n l a y e r i n t o the end

members, has n o t h i n g to do w i t h t h e q u e s t i o n which atom s p e c i e s i s a c t u a l l y d i f f u s i n g : i t

i s gove rned p u r e l y by t he o v e r a l l r e a c t i o n e q u a t i o n . A l t h o u g h t he p e n e t r a t i o n i s deepe r

towards t he t i n - r i c h s i d e , Ni i s t he f a s t e r d i f f u s i n g component .

b] A s i m p l e mass b a l a n c e can p r o v i d e g l o b a l i n f o r m a t i o n a b o u t t h e p o s i t i o n of t he K i r k e n d a l l

p l a n e d e p e n d i n g on t h e r a t i o of t he i n t r i n s i c d i f f u s i o n c o e f f i c i e n t s . A more a c c u r a t e

a n a l y s i s t h r o u g h Eq. (12a ) i s needed e s p e c i a l l y i f d i f f e r e n t p a r t i a l mo la r volumes and

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

c] The a c c u r a c y of t h e measured p o s i t i o n of t h e K i r k e n d a l l p l a n e has to be t remendous i n

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

t h i s i s l a r g e r t h a n a b o u t 5. Th i s has been d e m o n s t r a t e d by B a s t i n e t a l . [34] end i s shown

as an example f o r t h e above m e n t i o n e d Ni3Sn4/Ni3Sn c o u p l e i n F ig . 2 . 9 , where

s t o i c h i o m e t r i c compounds a r e assumed end Eq. (12a) i s used f o r t h e c a l c u l a t i o n of

DNi/Dsn , taking VNi/Vsn = 0.38. In fact, one must admit that the diffusion couple

technique is not suited for the accurate determination of the ratio between the intrinsic

diffusion coefficients if this is larger than about 5, even if the partial molar volumes

of the components are exactly known. For a layer width of 100 ~n, the position of the

markers for the cases DNi/Dsn = 5 and DNi/Dsn = ~ differs by 3.A ~m, which is mostly

within the accuracy limits of the experiment.

I0

DNi/Dsn

2

1 0.5 1 ~ 2 I ~ I ~ l!/l I

Fig . 2 . 9 . The r a t i o of t he d i s t a n c e s x ~ x I i n r e l a t i o n to t h e r a t i o o f t h e - i n t r i n s i c d i f f u s i o n c o e f f i c i e n t s DNi/Dsn.

66 F. J, J. van Loo

In t h e f o r e g o i n g i t was assumed t h a t t h e K i r k e n d a l l p l a n e can be f o u n d , e . g . by u s i n g

i n e r t makers . The a n a l y s i s showed, t h a t t h e p o s i t i o n of t h i s p l a n e has to be d e t e r m i n e d v e r y

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

c o e f f i c i e n t s . The m a r k e r s h a v e , t h e r e f o r e , to be v e r y t h i n . The b e s t method , a l t h o u g h i t

does n o t work i n a l l c a s e s , i s to use ' ~ a t u r a l " m a r k e r s which may be made v l s l b l e by a

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

e s p e c l a l l y a t p l a c e s where t h e c o n t a c t be tween t h e end members was n o t e s t a b l i s h e d

i m m e d i a t e l y , can t h e n o f t e n be s e e n a l o n g a s t r a i g h t row. O f t e n t h e g r a i n s t r u c t u r e of t h e

phase a t b o t h s i d e s of t h i s row i s d i f f e r e n t , which can e a s i l y be u n d e r s t o o d from e . g .

F ig . 2 . 8 . The p a r t of t he Ni3Sn 2 l a y e r above t h e K i r k e n d a l l p l a n e i s formed by t h e d i f f u s i o n

of atoms i n t o and o u t of Ni3Sn 4 , whereas t h e p a r t be low t h e F d r k e n d a l l p l a n e i s formed by

the d i f f u s i o n i n t o and o u t o f Ni3Sn. In some way t h e d i f f e r e n t g r a i n morphology o f Ni3Sn ~

and Ni3Sn i s r e f l e c t e d i n t h e d i f f e r e n t morphology i n t he Ni3Sn2-Phaae on b o t h s i d e s of t he

K i r k e n d a l l p l a n e . Nice examples ,my be found i n Ni -Sn [33]0 Nb-Sn [ 3 6 ] , end NIO-A1203 [37 ] .

The use of t h i c k m a r k e r s , l i k e e . g . t u n g s t e n w i r e s of 10 pm d i m , e r r i s v e r y dangerous

as was c o n v l n c i n g l y d e m o n s t r a t e d by B a s t l n [ 3 8 ] . F i g . 2 .10 shows t h e f a t e of an o r l g l n a U y

c y l l n d r l c a l t u n g s t e n w i r e i n a T i - N i d i f f u s i o n c o u p l e . S i n c e a f t e r a s h o r t d i f f u s i o n t ime

the t h i c k w i r e i s i n c o n t a c t w i t h a l l p h a s e s o f t h e c o u p l e , i n e a c h of which t he

d i f f u s i v i t i e s and v a c a n c y f l u x e s were d i f f e r e n t , an a c t u a l r u p t u r e o f t h e w i r e took p l a c e .

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

p a r t i c l e s r e v e a l e d t h e t r u e p o s i t i o n of t h e K i r k e n d a l l p l a n e . The same e f f e c t cou ld be seen

F ig . 2 .10 . The d i s t o r t i o n o f an i n i t i a l l y c i r c u l a r t u n g s t e n w i r e i n a d i f f u s i o n c o u p l e N i / T i a f t e r 72 h r a n n e a l i n g a t 800°C [ 3 8 ] .

Multipl~ase DiffuSion 6?

I I ////

cu y I zn

Fig . 2 .11 . The d i s t o r t i o n and d i s p l a c e m e n t of i n i t i a l l y s q u a r e m i c r o - i n d e n t a t i o n s in a Cu-Zn c o u p l e a f t e r ~ h r a n n e a l i n g a t 400"C [39 ] .

i f m l c r o h a r d n e s s i n d e n t a t i o n s were used as ma rke r s a s shown i n F i s . 2 .11 [ 3 9 ] . E l o n g a t i o n of

the i n d e n t a t i o n s o c c u r s , and i n f a c t from t h e o r e t i c a l c o n s i d e r a t i o n s , d e p e n d i n g on t he r a t i o

of t he i n t r i n s i c d i f f u s i o n c o e f f i c i e n t s i n t h e v a r i o u s p h a s e s , d i s r u p t u r e of the

i n d e n t a t i o n s m i g h t happen . In Ref . [39] i t i s a l s o shown t h a t i f t h e ma rke r s i n the

K i r k e n d a l l p l a n e a r e f i x e d to t h e " l a b o r a t o r y " f r a m e , t he ends of t he c o u p l e w i l l be

d i s p l a c e d w i t h r e s p e c t t o t h a t f r a m e . I f , on t he o t h e r hand , one end of t he coup le i s f i x e d

to t he l a b o r a t o r y f r a m e , t h e K i r k e n d a l l marke r s move w i t h r e s p e c t to t h i s f r ame . The f i n a l

morphology and t h i c k n e s s of t h e d i f f u s i o n zone i s t h e same f o r b o t h e x p e r i m e n t a l s e t u p s .

F i n a l l y , somet imes an e a s y method can be used t o d e t e r m i n e t h e r e l a t i v e d i f f u s i o n

v e l o c i t i e s of t h e e l e~seu t s . By growin$ a phase ¥ be tween t h e pure end members A and B, and

r e h e a t i n g t he same c o u p l e d u r i n g t h e same t ime a t t h e same t e , , p e r a t u r e a g a i n as shown i n

F ig . 2 . 1 2 , t he newly formed p a r t of t he l a y e r can o f t e n be r e v e a l e d by s p e c i f i c e t c h i n g

p r o c e d u r e s . I t i s now c l e a r t h a t o n l y A atoms d i f f u s e s i n c e t h e l a y e r grows a t t he ¥/B

i n t e r f a c e . The method was used e . g . i n t he Ni-A1 [~0] and t he Ti-A1 [1~] s y s t e m .

6~ F . J . J . van Loo

8

Y

A

(a) (b)

B

" r~f f~"

Y "otd"

A

Fig . 2 .12 . R e a n n e a l i n g o f a c o u p l e A/B i n which t h e y - p h a s e i s formed ( a ) l e a d s to new f o r m a t i o n o f ¥ o n l y a t t h e B - s i d e , showing A t o be t h e o n l y d i f f u s i n g component .

The r o l e of i n t e r o h a s e i n t e r f a c e 4n t h e c r e a t i o n o r mn,~JhJla t4on of v a ~ , , e i a e r e s u l t i n e f r . - -

t he K i r k e n d a l l e f f e c t

In r e c e n t p u b l i c a t i o n s [8 ,41 ] t he r o l e of I n t e r p h a s e i n t e r f a c e s i n a d i f f u s i o n c o u p l e on

t he c r e a t i o n on a n n i h i l a t i o n of K i r k e n d a l l v a c a n c i e s w u I n v e n t I s a t e d . As an example , t he

most s i m p l e m u l t l p h a s e c o u p l e , v l z . an ct/~ c o u p l e w i t h o n l y one i n t e r f a c e , was c h o s e n .

Dependent on t h e i n i t i a l c o n c e n t r a t i o n s o f t h e end members , t h e same i n t e r f a c e must b e a b l e

to a c t a s a s o u r c e ( F i S. 2 .13 a , b , e ) o r a s i n k ( F i s . 2 .13 d) f o r v a c a n c i e s i n t h e c a s e where

the same e l e m e n t d i f f u s e s f a s t e r i n b o t h p h a s e s . I f t h e i n i t i a l c o n c e n t r a t i o n s a r e chosen in

such a way t h a t t h e K i r k e n d a l l p l a n e c o i n c i d e s w i t h t h e p o s i t i o n o f t h e i n t e r p h a s e i n t e r f a c e

( F i g . 2 .13 c ) no d i s c o n t i n u i t y i n t h e v a c a n c y f l u x i s f o u n d .

I f t he r a t i o of t h e i n t r i n s i c d i f f u s i v i t i e s i s > 1 i n one phase b u t

I i n t h e O t h e r p h a s e , i n t e r e s t i n g phencqena o c c u r . In F i g . 2 . 1 4 t h e i n i t i a l c o n c e n t r a t i o n s

a r e chosen i n such a way t h a t t h e Matano p l a n e c o i n c i d e s w i t h t h e phase i n t e r f a c e . In

F ig . 2 .14 a ) B d i f f u s e s f a s t e r t h a n A i n t h e same r a t i o i n b o t h t h e ~ and ~ - p h a s e , a l t h o u g h

t h e i n t e r d i f f u s i o n c o e f f i c i e n t i n ~ i s l a r g e r t h a n i n ~ ; i n F i g . 2 .14 b ) B d i f f u s e s f a s t e r

t h a n A i n t h e ~ - p h a s e b u t s l o w e r ~n t he ~ - p h a s e , and i n F i g . 2 .14 c ) B d i f f u s e s f a s t e r t han

A i n t h e ~ - p h a s e b u t s l o w e r i n t h e ~ - p h a s e .

In c a s e a ) t h e i n t e r f a c e a b s o r b s v a c a n c i e s f rom t h e ~ - p h a s e and pomps more v a c a n c i e s i n t o

t he ~ - p h a s e , t h u s a c t i n g as a s o u r c e f o r v a c a n c i e s . In c a s e b ) t h e i n t e r f a c e a c t s a s a

s o u r c e , s e n d i n g v a c a n c i e s i n t o b o t h t h e ~ and t h e ~ - p h a s e . In r e f s . 8 end 41 i t has been

shown t h a t i n such a c a s e two r e a l K i r k e n d a l l p l a n e s have to be d e f i n e d . M a r k e r s , p l a c e d a t

the o r i g i n a l i n t e r f a c e a t t = 0 , have to s p l i t up and d r i f t i n b o t h t h e ~ and t h e ~ - p h a s e .

If t h e r a t i o DA/D B i s c o n s t a n t i n each p h a s e , t h i s p o s i t i o n c o i n c i d e s w i t h t he two

i n f l e c t i o n p o i n t s i n t h e p e n e t r a t i o n p l o t [~ ,&2] . In c a s e c) t he i n t e r f a c e a c t s as a s i n k

f o r v a c a n c i e s coming from b o t h t h e ~ and ~ - p h a s e . I t i s e a sy to imag ine t h a t t h e a r r i v a l of

v a c a n c i e s a t t h e i n t e r f a c e from b o t h s i d e s migh t g i v e r i s e t o t h e f o r m a t i o n of p o r e s , and

t h e r e f o r e , had a d h e r e n c e be tween t he end members. In t h i s c a se two ' ~ i r t u a l " K i r k e n d a l l

p l a n e s may be d e f i n e d , p o s i t i o n e d a t t he i n f l e c t i o n p o i n t s of t h e v i r t u a l e x t e n s i o n s of t he

p e n e t r a t i o n p l o t s i n b o t h p h a s e s . In c a s e a ) one r e a l and one v i r t u a l K i r k e n d a l l p l a n e

O c c u r s •

Muitiphase Diffusion 69

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M.Ka KP ÷1~

| M,,K o. Ki5

1

P

T XB

e) -Ib

ALpha

-~'0

o

"-'-~-- Jo : J,

~ Jv o

o

• i 1

-M KP +10

Beta

I-M KP

I o Io I o ,o °

-10 ~}a

31+

JA Jv

Beta

.1'0

M.K~ j tKO.

b)

d)

Fig . 2 .13 . P e n e t r a t i o ~ p l o t s of component B (mole f r a c t i o n ) i n a sys tem i n which D B = 10 D A i n b o t h t h e ~ and ~ phase f o r v a r i o u s i n i t i a l end member c o n c e n t r a t i o n s and

D: - I0 D: . A l so i s shown t h e v a c a n c y f l u x JV and i n a ) and c ) t h e g r a d i e n t of t h e v a c a n c y g lux as a f u n c t i o n of t h e p o s i t i o n i n t h e c o u p l e [ 8 ] .

70 F, J. J. van Loo

a)

1

£S

T XB

O.

ALpha

-~'0

:+____---- "" ++~-- J8

+" -+A 0

- ~ 3v ,.,.

l.M KP +~0

IoM KP

b)

!+

t 0

o ; o 0 ' 0

0 i 0 0 i 0 i 0 +

0

o : I~ o °B 0 | 0 0 J - o " I o - r o JA

° I o ° < ,, • 1 -u

o , ~ o i

-10 K = M-I KO +10

I , , I ~ M KP

I

I •

Atpha ~ " K= Beta

c) c' _,~ t . , -.+Co

~Vol' , _ . . ._ . . . . . .~M.t , [ X¢ =L~.._...---~

F i g . 2 . 1 4 . P l o t s o f t h e mole f r a c t i o n o f B and t h e v a c a n c y f l u x 3V i n c o u p l e s whe re the Matano i n t e r f a c e does n o t move, I n ( a ) DB/D A = l0 i n b o t h t h e = - and ~ - p h a s e s , i n <b) D ~ / D , - 10 i , the ~ - p ~ . . and Oil " i , th , =-phase. and in ~o~ D , / D , - 0.1 in t h e ~ - p h a s e and 10 i n t h e = - p h a s e [

Multiphase Diffusion '71

These phenomena may be v i s u a l i z e d by t he b e h a v i o u r of an i n i t i a l l y s q u a r e i n d e n t a t i o n a t

the i n t e r f a c e a t t = 0 as shown i n F i g . 2 .15 .

The same a u t h o r s p r e s e n t e d a model f o r t he dynamic a c t i o n of t h e i n t e r f a c e d u r i n g the

phase t r a n s f o r m a t i o n , which i n c l u d e s c l imb of m i s f i t d i s l o c a t i o n s o u t of t h e i n t e r f a c e and

c l imb of m i s o r i e n t a t i o n d i s l o c a t i o n s i n t he i n t e r f a c e [ 4 3 ] .

Beta

-

t=O 4b Eb ?b Fllpha

- H = I

Fig . 2 .15 . The change i n morphology of a s q u a r e i n d e n t a t i o n m a r k e r , o r i g i n a l l y p l a c e d a t the ~ /~ i n t e r f a c e a t t = 0 a f t e r d i f f u s i o n a n n e a l i n g i n t he s i t u a t i o n s g i v e n by f i g s . 2 . 1 4 a - c . The r e a l o r v i r t u a l d i s p l a c e d o r i g i n of t~e a l p h a and b e t a K i r k e n d a l l f r ames of r e f e r e n c e a r e d e s c r i b e d by 1~ and K ~ ( d e f i n e d by t h e c o n s t a n t l a t e r a l i n d e n t a t i o n w i d t h ) [ 8 ] .

The morvholo~v of i n t e r o h a s e i n t e r f a c e s i f s t a i n b o u n d a r v d i f f u s i o n i s t a k i n e , l a c e

I f a s l n g l e - c r y s t a 1 1 i n e l a y e r of 7 i s growing be tween s l n g l e - c r y s t a l l l n e s u b s t r a t e s = and

~, one migh t e x p e c t c o m p l e t e l y s t r a i g h t = / 7 and y / ~ i n t e r f a c e s , p r o v i d e d t h a t no o r i e n t a t i o n

r e l a t i o n e f f e c t s o c c u r which m i g h t cause a f a c e t e d i n t e r f a c e . In p o l y c r y e t a l l i n e l a y e r s

grown be tween p o l y c r y s t a l l i n e end members, t he i n t e r f a c e s a r e o f t e n n o t c o m p l e t e l y s t r a i g h t .

Some t y p i c a l s t r u c t u r e s a r e g i v e n i n F i g . 2 .16 .

In many d i f f u s i o n e x p e r i m e n t s , c a se a ) i s found : an e q u i a x e d l a y e r of ¥ bounded by

c o m p l e t e l y s t r a i g h t i n t e r f a c e s . Of t en t h i n n e e d l e - l i k e c r y s t a l s a r e formed, e . g . Cu3Si

f o r m a t i o n i n Cu-Si c o u p l e s [ 1 6 , 4 4 , 4 5 ] as shown i n c a s e b ) where s t r a i g h t i n t e r f a c e s a r e

found . The e x p l a n a t i o n i s t h a t i n c a s e a ) o n l y volume d i f f u s i o n o c c u r s , e . g . b e c a u s e of the

use of r e l a t i v e l y h i g h t e m p e r a t u r e s so t h a t g r a i n - b o u n d a r y d i f f u s i o n becomes u n i m p o r t a n t . In

ca se b ) , on t h e o t h e r hand , o n l y g r a i n - b o u n d a r y d i f f u s i o n t a k e s p l a c e . The n e c e s s a r y l a t e r a l

d i f f u s i o n a t t h e i n t e r f a c e s m i g h t o c c u r by s u r f a c e o r volume d i f f u s i o n , b u t i n f a c t t h i s i s

i m m a t e r i a l b e c a u s e of t h e t h i n n e s s of t h e c r y s t a l s . In c a s e c ) i n t e r d i f f u s i o n t h r o u g h

g r a l n - b o u n d a r i e s i s f a s t e r t h a n i n t e r d i f f u s i o n t h r o u g h t he b u l k as e . g . i n t he Ni3A1 phase

grown i n Ni/NiA1 c o u p l e s [ 4 0 ] . The morphology g i v e s no i n f o r m a t i o n a b o u t the r e l a t i v e

m o b i l i t y of t he two componen ts .

In ca se d ) , found e . g . a t t he FeSn2/FeSn i n t e r f a c e i n a FeSn2/Fe c o u p l e [33] and in a much

more e x a g g e r a t e d form e . g . t he f o r m a t i o n of Fe2A15 [46] b u l k d i f f u s i o n i s p r e d o m i n a n t , b u t

i s d i f f e r e n t f o r v a r i o u s c r y s t a l l o g r a p h i c d i r e c t i o n s which l e a d s to a d i f f e r e n t l e n g t h of

the g r a i n s . A n o n - u n l f o r m n u c l e a t i o n accompanied by l a t e r a l growth may a l s o l e ad to

morphologies like this [44].

72 F.J .J . van Loo

Illllltlllllllrl}tl]llilll111}llllllllllllllllll cb'

(e)

(d)

(f)

Fig . 2 .16 . V a r i o u s m o r p h o l o g i e s of a r e a c t i o n l a y e r i n a b i n a r y d i f f u s i o n c o u p l e , e x p l a i n e d i n t h e t e x t .

Gra in b o u n d a r i e s i n t h e end members may a l s o a f f e c t t h e morphology of t h e r e a c t i o n l a y e r

a t r e l a t i v e l y low t e m p e r a t u r e , e s p e c i a l l y u n d e r t h e i n f l u e n c e of i m p u r i t i e s ( F i g . 2 .16 e , f ) .

For i n s t a n c e , B a s t i n [34] found e x t e n s i v e f o r m a t i o n of ~ - T i by g r a i n - b o u n d a r y d i f f u s i o n of

Ni i n t o m-Ti i n t h e p r e s e n c e of t r a c e s of i r o n . I r o n a t a b i l i s e 8 ~ -T i i n t he g r a i n - b o u n d a r i e s

of m-Ti and , t h e r e f o r e , e n h a n c e s N i - d i f f u s i o n . The removal of i r o n s t o p p e d t h i s t y p e of

g r a i n - b o u n d a r y i n t e r d i f f u s i o n . Whether morphology e ) o r f ) i s found depends on t h e p o s i t i o n

of t he r e l e v a n t i n t e r f a c e w i t h r e s p e c t to t h e Matano p l a n e : t h e t o t a l l a y e r w i d t h i n c r e a s e s

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

Matano p l a n e . Morphology f ) can a l s o be found i f e . g . t h e f o r m a t i o n of a s o l i d s o l u t i o n i n

t he end member i s f a c i l i t a t e d by ( i m p u r i t i e s i n ) t h e g r a i n bounda ry [ 1 4 ] .

I n f l u e n c e of a d i f f e r e n c e i n c r o s s - s e c t i o n of t h e end members on t h e l a v e r morDholoev i n

Dur ing r e s e a r c h on i n t e r d i f f u s i n n of Cu and Si [65] a t y p i c a l phenomenon was found , as

s c h e m a t i c a l l y shown i n F i g . 2 . 1 7 .

I f a s m a l l p i e c e o f S i i s u s e d , t h e C u 3 S i - l a y e r grows i n e x a c t l y t h e same c r o s s - s e c t i o n i n t o

t he Cu end member. I f t he S i ha s a l a r g e r c r o s s - s e c t i o n t h a n Cu, a l e n s - s h a p e d l a y e r i s

formed, b r o a d e r t h a n t h e o r i g i n a l Cu end member. In b o t h c a s e s t h e co lumnar g r a i n s of Cu3Si

a lways grow p e r p e n d i c u l a r to t h e o r i g i n a l i n t e r f a c e .

The e x p l a n a t i o n has to be found i n t h e f a c t , t h a t Cu i s t he o n l y d i f f u s i n g component i n

Cu3Si. In t h e l e f t - h a n d f i g u r e , Cu atoms d i f f u s e s t r a i g h t t h r o u g h Cu3Si to the a v a i l a b l e S i ,

c o n v e r t i n g t h e o r i g i n a l S i i n t o a Cu3Si mass of t he same shape ( a l t h o u g h w i t h a volume of

a b o u t 7 /3 t£mes t he volume of t h e r e a c t e d S i ) . In the r i g h t - h a n d f i g u r e , however , Cu can

d i f f u s e by s u r f a c e d i f f u s i o n a l o n g t he Cu3Si to r e a c t w i t h Si a v a i l a b l e o u t s i d e t he o r i g i n a l

c r o s s - s e c t i o n of Cu, l e a d i n g to t h e l e n s - s h a p e d morphology of t he r e a c t i o n l a y e r .

C o n v e r s e l y , t h e f a s t e r d i f f u s i n g component can be i d e n t i f i e d from t h i s morphology .


Cu

IIIIIiiii 1rll flail Illliliiiillll

$i 5i

Fig. 2.17. D i f f e r e n t uorphologies of the C u ~ i r eac t i on l aye r in a Cu/Si d i f f u s i o n couple dependent on the r e l a t i v e d i ~ e t e r s of the end members. The columnar gra ins in Cu3Si are i nd i ca t ed .

Genera l ly , the edges and f r e e sur faces of the d i f f u s i o n couple g ive ex t r a information on

the l aye r growth mechanian and concen t ra t ion l i m i t s of the var ious phases. Often surface

d i f f u s i o n or evapora t ion /condensa t ion processes g ive r i s e to the formation of a l a rge r

~ t og phases r i c h in s u b s t r a t e ma te r i a l compared to the r e ~ l a r d i f f u s i o n zene because

of an impeded supply of d i f f u s i n g ma te r i a l (see c-phase in Fig . 2.18) In f a c t , a kind of

incremental couple i s p resent a t these edges. In t e rna ry d i f f u s i o n couples , t h i s phenomenon

can be used to even more advantage (see chapter 3).

8

A

Fig. 2.18. Development of d i f f u s i o n l aye r s a t the edges of a d i f f u s i o n couple. ~:14 r

74 F.J.J. van Loo

Comvar ison o f t he r e a c t i o n l a v e r t h i c k n e s s i n o l a n a r , c y l i n d r i c a l and a o h e r i c a l d i f f t m i o n

CQuDles

I t makes a d i f f e r e n c e when l a y e r g rowth i s s t u d i e d i n p l a n a r d i f f u s i o n c o u p l e s a s

d i s c u s s e d up to now o r i n e . g . c y l i n d r i c a l o r s p h e r i c a l c o u p l e s , wh ich a r e i m p o r t a n t i n e . g .

c o a t e d w i r e s , f i b r o u s c o m p o s i t e s o r i n s i n t e r i n g p r o b l e m s . Le t us look to t h e g rowth o f a

l a y e r &By be tween an i n f i n i t e m a t r i x B and pu re A which h a s t h e d i m e n s i o n s o f e i t h e r a p l a t e

w i t h t h i c k n e s s 2 r o , o r a c y l i n d e r w i t h d i a m e t e r 2 r o , o r e l s e a s p h e r e o f t he same

d i a m e t e r . The volume o f one s o l e &By i s z t i m e s t he voluDe o f one mole o f pu re A. A f t e r some

t ime t , t h e S i t u a t i o n shown in F i g . 2 .19 a r i s e s , where a e q u a l s t h e u n r e a c t e d r a d i u s , o r

( f o r t h e p l a t e ) a e q u a l s h a l f t h e u n r e a c t e d t h i c k n e s s o f A.

(a )

A 3y

I ro

I v'-

t I

A

I I I I I

( b )

-I I

A By B

Fig . 2 . 1 9 . The r e a c t i o n be tween a p l a t e o f pu re A w i t h t h i c k n e s s 2 r o ( a ) and of a s p h e r e or c y l i n d e r of pu re A w i t h d i a m e t e r 2 r o (b) w i t h a m a t r i x B, f o r m i n g t he compound ABy of l a y e r t h i c k n e s s d .

The c o n v e r s i o n f a c t o r m, i . e . t h e f r a c t i o n of A which h a s r e a c t e d , i s :

r - a o a ( ]Pla te) == ; =I-~ r r o o

2 2 r - a 2

g L_ = 1 - ~ ( C y l i n d e r ) ~ = 2 ; 2

r r o o

3 3 r - a 3

0 ~L. = 1 - ~ (Sp h e re ) ~ = 3 ; 3

r r o o

(2~)


From g e o m e t r i c a l c o n s i d e r a t i o n s the f o l l o w i n g r e l a t i o n s e x i s t between = and the l a y e r

t h i c k n e s s d:

d p / r o - z

d c l r o = {i + (s - I )~} I12 - ( I - z ) l 1 2

d s / r o = {I + (z - I ) ~ } I 1 3 - ( I - a) 113

(Plate) 1

(Cylinder) (25)

(Sphere)

We a re i n t e r e s t e d in = and d as a f u n c t i o n of t ime i f volume d i f f u s i o n i s the

r a t e - l i m i t i n g s t e p . As an example, c o n s i d e r the fo rma t ion of an ABy- l aye r , which grows in a

p l a n a r couple a c c o r d i n g to the p a r a b o l i c law d 2 = 2 kpt ( ~ = p a r a b o l i c growth c o n s t a n t ) .

For s h o r t a n n e a l i n g t i m e s , the l a y e r wid ths a r e equa l f o r the t h r e e m o r p h o l o g i e s , bu t f o r

l o n g e r a n n e a l i n g , d i f f e r e n c e s a r i s e from the f a c t t h a t t he s u r f a c e a r e a of the i n n e r and

o u t e r s h e l l of the l a y e r w i l l change, c o n t r a r y to the c o n s t a n t s u r f a c e in p l a n a r c o u p l e s .

A number of e q u a t i o n s a r e g iven in the l i t e r a t u r e in v a r i o u s deg rees o f approx imat ion and

v a l u e s of z [47-52] . The a u t h o r has nowhere found a c l e a r compar i son , e x p r e s s e d in the same

quantles and, therefore, a number of useful relations is given here between the values of =,

ro , t and the p l a n a r p a r a b o l i c growth c o n s t a n t kp. The d e r i v a t i o n of t h e s e e q u a t i o n s

proceeds a long the same l i n e s as used by C a r t e r [47] , bu t s p e c i a l c a r e i s taken in d e f i n i n g

the v a r i o u s q u a n t i t i e s which l e ads to e x p r e s s i o n s s l i g h t l y d i f f e r e n t from those g iven in the

l i t e r a t u r e :

(z~) 2 = 2 k p t l r ~ P l a t e --

"2 [ , In

2 - z~l(l+(z-1)~}213-s(1-~)213=2kp t i t - ~ o Sphere - -

z -1

z - 1: (1-~13)-(Z-~)213-2kptlr2 o

For low v a l u e s o f a t he Zqs. (25 ) and (26 ) reduce t o :

(z~)2=d21r2=2k t / r 2 P l a t e ~- - p o p o /

(z~)2=4d21r2=Sk t l r 2 ~ (27) c o p o

(z~)2=9d2/r2=18k t / r 2 Sphere s o p o

Cyl inde r

C y l i n d e r

(26)

By p l o t t i n g the c o n v e r s i o n f a c t o r ~ v e r s u s t ~ a c c o r d i n g to Eqs. (26) f o r z = 2 as shown

in Fig. 2.20, straight lines are found with slopes of I, 2 and 3 times (2kp)~/Zro for values

of = lower than about 0.35. Therefore, from each slope the true value for the parabolic rate

constant of a planar couple can be found.


0.5

0 ' / 0 5 10 15 20

- tV2(a. .)

Fig . 2 .20 . A p l o t o f t he c o n v e r s i o n f a c t o r ~ v e r s u s t ~ f o r a p l a t e of t h i c k n e s s 200 ( p ) , a c y l i n d e r ( c ) and a s p h e r e ( s ) b o t h o f d i a m e t e r 200 of m a t e r i a l A r e a c t i n g wi th a matrix B according to A + B ~ AB. Volume diffusion is the rate limiting step. The volume r a t i o AB/A = 2, the p a r a b o l i c growth c o n s t a n t f o r t he p l a t e kp = 50, a l l i n a r b i t r a r y u n i t s .

Through Eq. (25) t he a / t ~ p l o t can be c o n v e r t e d i n t o a d / t ~ p l o t as shown in F i g . 2.21

for s - 2. Consistent with the foregoing result, up to a thickness of about 0.35 r o the

t h i c k n e s s e s a r e n e a r l y equa l f o r t he t h r e e coup le t y p e s . I t t a k e s t h i c k n e s s v a l u e s up to

about 1.1 r o (= = 0 .95 ) f o r a c y l i n d e r and 0 .7 r O (= m 0 .90 ) f o r a s p h e r e to f i n d an

i n c r e a s e of 10% in l a y e r w i d t h compared w i t h a p l a n a r c o u p l e . For lower v a l u e s of s , the

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

However, in practice, the deviations are probably much larger than predicted from these

geometrical relations. One reason has bean pointed out by Beke [53]: diffusion into a

cylinder or sphere may build up stresses which cannot be released as easily as in a planar

c o u p l e s . These s t r e s s e s may i n f l u e n c e the t h i c k n e s s of the growing l a y e r ( s ) , e s p e c i a l l y when

vo l , - - e changes a r e i n v o l v e d . Also the K i r k e n d a l l e f f e c t can cause l a r g e d e v i a t i o n s ,

e s p e c i a l l y when the c y l i n d e r o r s p h e r e m a t e r i a l i s d i f f u s i n g much f a s t e r than the

s u r r o u n d i n g m a t e r i a l . In t h i s c a s e , a l a r g e i n c r e a s e of t he t o t a l volume may occur due to

the f o r m a t i o n o f ho l low s p h e r e s [5~] .


200

150

d (..u.)

1 IO0

5O

0 0

P

C

S

5 10 15 20 tl~(a.u.)

Fig. 2 .21. A p l o t of the t h i c k n e s s d of the AB r e a c t i o n l a y e r v e r s u s t ~ f o r the s i t u a t i o n d e s c r i b e d in Fig. 2 . 2 0 .

Texture in r e a c t i o n l a y e r s

Very o f t e n the c r y s t a l s of the phases In d l f fu s lo n - s r c r~ n r e a c t i o n l a y e r s have a p r e f e r r e d

o r i e n t a t i o n in the srowth ( d i f f u s i o n ) d i r e c t i o n . This t e x t u r e has been s t u d i e d e x t e n s i v e l y

in our l a b o r a t o r y and the f o l l o w i n s r e s u l t i s found: the d i r e c t i o n of the l o n s e s t

c r y s t a l l o s r a p h i c ax la i s i n v a r i a b l y o r i e n t e d p e r p e n d i c u l a r to the d i f f u s i o n d i r e c t i o n , i . e .

p a r a l l e l to the i n t e r f a c e [55 ,56] . Very o f t e n a n o t h e r r u l e of thumb 4s found: the s h o r t e s t

r e p e a t - d l s t a n c e in the c r y s t a l l a t t l c e i s a l i g n e d p a r a l l e l to the d i r e c t i o n of d l f f u a l o n

( f i b e r t e x t u r e ) . These t e x t u r e s a re most pronounced a t the srowth f r o n t of the r e a c t i o n

l a y e r ; in the " o l d e s t " p a r t of the l a y e r sometimes changes in t e x t u r e a r e found due to

r e c r y s t a l l i z a t i o n .

78 F.J .J . van Loo

The p r e s e n c e of a t e x t u r e i s i m p o r t a n t s i n c e o f t e n c h e m i c a l and m e c h a n i c a l p r o p e r t i e s a r e

q u i t e d e p e n d e n t on t h e c r y s t a l l o g r a p h i c d i r e c t i o n . In t h e we l l -known g a l v a n i z i n g p r o c e s s ,

f o r i n s t a n c e , t h e S-FeZnl0 l a y e r i s formed w i t h i t s h e x a g o n a l a x i s p e r p e n d i c u l a r to t he

d i f f u s i o n d i r e c t i o n [ 5 7 , 5 8 ] . In t h a t d i r e c t i o n t he d i f f u s i o n o f Zn t h r o u s h S i s v e r y f a s t

which c a u s e s r e l a t i v e l y t h i c k r e a c t i o n l a y e r s . For economic r e a s o n s i t would be v e r y h e l p f u l

i f t he h e x a g o n a l a x i s c o u l d be f o r c e d to o r i e n t i t s e l f p a r a l l e l t o t h e d i f f u s i o n d i r e c t i o n .

U n f o r t u n a t e l y , d e s p i t e t h e v a r i o u s t r i c k s we u s e d , n a t u r e d i d n o t a l l o w us to i n t e r f e r e w i th

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

The e v a l u a t i o n of d i f f u s i o n c o e f f i c i e n t s i n c o u p l e s i n which l a r e s volume ehan~as o c c u r

E q u a t i o n s (3 ) and (12) can be used w i t h o u t any problem f o r f i n d i n g t he ( i n t e r ) d i f f u s i o n

c o e f f i c i e n t s f rom a p e n e t r a t i o n c u r v e i n m u l t i p h a s e c o u p l e s , even i f v e r y l a r g e volume

changes o c c u r . A s p e c i a l example i s t h e g rowth of a t i t a n i u m c a r b i d e l a y e r be tween pure

t i t a n i u m and c a r b o n [ 1 0 ] . The p a r t i a l mo la r volume of c a r b o n i n t i t a n i u m c a r b i d e i s

v i r t u a l l y z e r o , which l e a d s to e q u a l v a l u e s f o r t he i n t e r d i f f u s i o n c o e f f i c i e n t D and t he

i n t r i n s i c d i f f u s i o n c o e f f i c i e n t of c a r b o n D C i n t i t a n i u m c a r b i d e :

= CTi VTi DC + CC 'Vc DTi = DC (lOa)

~he e v a l t t a t i o n of d i f f u s i o n c o e f f i c i e n t s i n v a o o u r - s o l i d c o u n l e s compared w i t h s o l i d - s o l i d

~oup les

Let us compare , f o r i n s t a n c e , t h e g rowth of an FeO l a y e r be tween Fe end Fe304 on t he one ~

hand , and be twe en Fe and oxygen gas w i t h an oxygen a c t i v i t y e q u a l to t h e Fe0/Fe304

e q u i l i b r i u m on t h e o t h e r hand [ 1 3 ] . The t h i c k n e s s of t h e F e O - l a y e r w i l l be l a r g e r i n the

f i r s t c a s e t h a n i n t h e l a t t e r b e c a u s e of t h e f a c t , t h a t FeO grows from two s i d e s : t he

r e a c t i o n of oxygen w i t h i r o n and t h e l o s s of oxygen from Fe304. The p e n e t r a t i o n c u r v e s ,

e x p r e s s e d i n mole f r a c t i o n s of i r o n , a r e s c h e m a t i c a l l y shown i n F i g . 2 .22 . For s i m p l i c i t y ,

t he homogene i ty r a n g e i n FeO i s t a k e n as z e r o , which i s o n l y a rough a p p r o x i m a t i o n b u t n o t

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

Accord ing to W a g n e r ' s a n a l y s i s [13] t h e i n t e g r a t e d d i f f u s i o n c o e f f i c i e n t f o r t he

FeO-phase can be w r i t t e n f o r t h e Fe /Fe30 A c o u p l e a s :

( l - O . S ) CO.S-~/7) d ~ / 2 t = Int = 4/7

For the Fe/oxygen couple one finds:

I-~6" d~/2t

(I-0.5)(0.5-0), d~/2t = ~. d2/2t Bint = (I-0)

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

d 1 = 2d 2. In t h e s e s i m p l e c a s e s ( o n l y l i ne - compounds a r e i n v o l v e d ) t h e v a r i a t i o n s i n p a r t i a l

molar volumes a r e i r r e l e v a n t and a p l o t w i t h m o l a r f r a c t i o n can be u sed .


I f the mole f r a c t i o n of oxygen had been used f o r the p l o t s , the same r e s u l t s must be

obtained. This means that for the oxygen gas the mole fraction must be taken as N O = I,

i r r e s p e c t i v e of i t s p r e s s u r e or a c t i v i t y . D i f f e r e n c e s in oxygen a c t i v i t y on ly r e f l e c t

themselves through differences in the defect structure in changes in the interdiffuslon

c o e f f i c i e n t f o r t he FeO-phase .

1

0.8

0.6

0.~.

0.2

Fe3Oz FeO

d I

Fe Fe

02

FeO ----4W

Ca) (b)

Fig . 2 .22 . The f o r m a t i o n of a FeO- laye r i n a d i f f u s i o n couple F e / F e ~ 6 (a) and in an o x i d a t i o n e x p e r i m e n t of Fe in an oxygen a tmosphere equa l to t he FeO/Fe30 A e q u i l i b r i u m ( b ) , a n n e a l e d f o r t he same p e r i o d of t ime .

The same p r i n c i p l e can be used i f d i f f u s i o n i s s t u d i e d between the a l l o y s AqBI_ q and

AyBl_y, h e a t e d in a c l o s e d v e s s e l w i t h no d i r e c t c o n t a c t . D i f f u s i o n then has to proceed

through the gas phase . In t h e i d e a l c a s e , t he same c o n s t a n t s u r f a c e c o n c e n t r a t i o n N A = p

(q<p<y) has to occu r in t he e q u i l i b r i u m c o n d i t i o n on the o u t e r s u r f a c e of b o t h a l l o y s . In

p r a c t i c e , sometimes a d i f f e r e n t bu t c o n s t a n t v a l u e i s found f o r b o t h s u r f a c e s [59-61] ,

showing an activity gap in the gas phase.

In F ig . 2.23 i t i s assumed t h a t on ly e lement A e v a p o r a t e s , and t h a t the s u r f a c e

concentration on both alloys equals N A = p. The alloy AyBl_y has lost material (thickness

d e c r e a s e x 0 - x 1) and AqBI_ q has ga ined m a t e r i a l ( t h i c k n e s s i n c r e a s e x 2 - X o ' ) . I f the molar

volumes of A and B a r e e q u a l , and i f the t o t a l s u r f a c e of b o t h a l l o y s i s e q u a l , then 0

x 0 - x 1 = x 2 - x o ' - The t o t a l amount of A, l o s t from the a l l o y AyBl_y, eq u a l s the sum of t he a r e a C and D. On

the o t h e r hand, i t a l s o e q u a l s the sum of the a r e a D and E. T h e r e f o r e , one may f i n d the

o r i g i n a l s u r f a c e x o by e q u a t i n g

Area C = Area E.


In the same way the &mount of component A g a i n ed by the AqBI_ q a l l o y e q u a l s t he a t e o f

the a r e a C + K + H, b u t a l s o e q u a l s K + G ÷ F, from which i t f o l l o w s t h a t t h e o r i g ~ w

s u f a c e x o i s £ound by e q u a t i n g

Area F ,, Area H

t

Under t he above men t ioned c o n d i t i o n s i n which x 0 - x I = x 2 - x 0 , one m i s h t s h i f t t he t

couple h a l v e s i n such a way t h a t x 0 = x 0 and x 1 = x 2. One than s e t s F ig . 2 .2~ , s h o w i n | t h ~

Area C + F - g = Area H

!

and, t h e r e f o r e , the p l a n e x 0 = x 0 i s i d e n t i c a l to the Hatano p l a n e .

~NA t s

Ay Bl.y

(a)

"'1 I

E: i I

- 4

0

I

x:~i . . j x x 0

pl F

q K'~ J

I

x 2 x 6

(b]

i

t"A i I

Xl:X 2 x0:x ~

F ig . 2 .23 . The a n n e a l i n g of t he a l l o y s A ~ l _ y and AqBI_ a t o g e t h e r in a c l o s e d v e s s e l . A f t e r e q u i l i b r i u m i s r e a c h e d , t h e same s u r f a c e c o n g e n t r a t i o n N a - p i s p r e s e n t oqz b o t h a l l o y s . The p o s i t £ o n s o f t h e Mateno p l a n e s x o and x o ' a r e i n d i c a t e d f o r b o t h a l l o y s , t h e y c o i n c i d e w i t h t h e o r i g i n a l s u r f a c e s . ~ e new s u r f a c e s a f t e r d i g £ u s i o n a n n e a l i n g a r e g i v e n by x 1 and x 2, r e s p e c t i v e l y .

F ig . 2 .24 . The s i t u a t i o n in a pure s o l i d - s t a t e d i f f u s i o n coup le A y B l _ ~ A ~ l _ o ~ o b t a i n e d by o v e r l a p p i n g of F i g s . 2 .23 a ) and b ) u n t i l the coamon c o ~ c e n c r a t ~ o n p .


From t h i s a n a l y s i s the f o l l o w i n g c o n c l u s i o n s can be drawn:

1) S ince the d i f f u s i o n c o e f f i c i e n t , c a l c u l a t e d from the p r o f i l e s i n F i g s . 2.23 and 2.2~ has

to be the same f o r each c o n c e n t r a t i o n i n d e p e n d e n t o f t he t o t a l amounts of t he a l l o y s i t f

f o l l o w s t h a t in F ig . 2.23 b the Matano p l ane has to be the p l ane x 0 . This p o s i t i o n can

be found g e o m e t r i c a l l y w i t h o u t any dependence on the a c t u a l c o m p o s i t i o n of the mas t e r

a l l o y AyBl_y by e q u a t i n g the a r e a s F and H. I f the mola r volumes of A and B a r e no t

e q u a l , c o r r e c t i o n s have to be made as d i s c u s s e d p r e v i o u s l y .

2) From the d i f f u s i o n p r o f i l e shown in F ig . 2.23 a , i . e . the m a t e r i a l - l o s i n g a l l o y , the

Matano p l ane (now s i t u a t e d o u t s i d e the a l l o y ) can be found in such a way t h a t

Area C = Area E. I t would l ead to c o m p l e t e l y wrong r e s u l t s i f one would d e f i n e the Natano

p l ane as t he p l ane x3, d i v i d i n g the c o n c e n t r a t i o n p r o f i l e in t he a l l o y in two p a r t s equal

in a r e a ( x 3 - x l - P - r = r - s - t ) .

3) I f b o t h A and B a r e e v a p o r a t i n g , o r i f t he t o t a l s u r f a c e of one a l l o y i s much l a r g e r than

the o t h e r ( e . g . a p l a t e of AqBI_ q embedded in a l a r g e e x c e s s of AyBl_y p a r t i c l e s ) a same

kind of a n a l y s i s a p p l i e s p r o v i d e d t h a t i n f i n i t e d i f f u s i o n co u p l e s s t i l l can he assumed.

In the f i r s t c a s e , the n e t d i f f e r e n c e (A-B) has to be equa l to Area D+E = Area K+G+F.

This means t h a t an equa l t r a n p o r t o f A and B l e a d s to a Matano p l ane c o i n c i d i n g wi th the

o u t e r s u r f a c e s of b o t h ~ l t o y ~ .

In the second c a s e , the c o n c e n t r a t i o n p a t t he o u t e r s u r f a c e would be much c l o s e r to the t

value y, making x0-x I < x2-x o . The two penetration curves can then not be joined to one

continuous curve llke in Fig. 2.2~: two different Matano planes can be defined, one for

each alloy.

A discontinuity in the slope will arise at the outer surfaces in such a way, that the

same diffusion coefficient is found whether the one or the other alloy is evaluated at

their outer surface concentration.

In f a c t , t he message i s s i m p l e : i f d i f f u s i o n c o e f f i c i e n t s have to be c a l c u l a t e d from a

p e n e t r a t i o n c u r v e , t he ma thema t i ca l p o s i t i o n of t he Matano p l ane has to c o i n c i d e w i t h the

p o s i t i o n of t he o r i g i n a l s u r f a c e , which may be s i t u a t e d i n s i d e or o u t s i d e the a l l o y . From an

extension of these considerations interesting conclusions can be drawn concerning the

position of, and the composition at the Kirkendall plane [42]. In the specific example

chosen here, this plane coincides with x I at N A = p in Fig. 2.2A.

3. MULTIPHASE DIFFUSION IN TERNARY SYSTEMS

In the i n t r o d u c t i o n the i m p o r t a n t d i f f e r e n c e between b i n a r y and t e r n a r y sys tems was

a l r e a d y men t ioned . The e x t r a deg ree of f reedom in a t e r n a r y sys tem g i v e s r i s e to the

o c c u r r e n c e of two-phased l a y e r s in d i f f u s i o n c o u p l e s , end e s p e e l a l l y the r e l a t i o n between

the c o m p o s l t i o n of t he d i f f u s i o n zone and the phase diagram becomes much more c o m p l i c a t e d .

In a c l a s s l c a l p a p e r , K i rka ldy and Brown [62] f o r m u l a t e d a number or r u l e s which r e l a t e

t he c o m p o s i t i o n of t he d i f f u s i o n zone w i t h t he phase d iagram. They used the concep t of the

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

the ave rage c o m p o s i t i o n s in p l a n e s p a r a l l e l to t he o r i g i n a l i n t e r f a c e t h roughou t the

d i f f u s i o n zone .

82 F, J. J. van Loo

/ - - I - . ~ [-:-- %,--X B X

T C

A

(a)

C

} X

T C

(b)

A B X XlT C

C

A B X

0 o o o

° °T° c q.,o

oz-porec, o 0

o o

(c) (d)

Fig . 3 .1 . Examples o f p o s s i b l e m o r p h o l o g i e s f o r t he r e a c t i o n l a y e r i n a d i f f u s i o n coup le X/C. The c o r r e s p o n d i n g d i f f u s i o n p a t h s a r e p l o t t e d on t he i s o t h e r m s [65 ] .


In F i g . 3 .1 a h y p o t h e t i c a l phase d iag ram shows t h e e x i s t e n c e of s i x s i n g l e - p h a s e r e $ i o n s ,

v i z . t h e t h r e e e l e m e n t s A, B and C, two b i n a r y compounds X and Z, and t he t e r n a r y ccQpound

T. In F i g u r e 3 .1 f o u r d i f f u s i o n p a t h s a r e g i v e n , t o g e t h e r w i t h t h e c o r r e s p o n d i n g

m o r p h o l o g i e s of t he d i f f u s i o n zone , f o r t h e d i f f u s i o n coup le C v e r s u s X. They a l l f u l f i l l

t h e most i m p o r t a n t c o n s t r a i n t , v i z . t he c o n s e r v a t i o n of mass : no m a t e r i a l can be l o s t o r

c r e a t e d . Th i s f o r c e s t h e p a t h to c r o s s t he s t r a i g h t c o n n e c t i o n l i n e be tween C and X a t l e a s t

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

l i k e T and Z i n F ig . 3 . l b . O b v i o u s l y , t he Z -phase has to be much t h i n n e r t h a n t h e T-phase in

o r d e r to comply w i t h t h e mass b a l a n c e .

Not o n l y t he r e l a t i v e t h i c k n e s s e s , b u t a l s o t he t o t a l t h i c k n e s s of t he d i f f u s i o n l a y e r i s

r e l a t e d to t he d i f f u s i o n p a t h . I f e . g . i n t he Z -phase t he d i f f u s i o n i s v e r y s low and i n t he

T-phase f a s t , t h e n o b v i o u s l y t he t o t a l l a y e r w i d t h f o r F i g s . 3 . 1 . b o r c w i l l be s m a l l e r than

i n t h e ca se of F ig . 3 . 1 . d . In t h e f i r s t examples , t he c o n t i n u o u s Z - l a y e r a c t s as a k ind of

b a r r i e r , whereas i n t he c a s e of F ig . 3 . 1 . d , t he t o t a l l a y e r t h i c k n e s s i s governed by the

f a s t e r d i f f u s i o n i n t he T - p h a s e , which i s p r o b a b l y o n l y l i t t l e hampered by t he d i s c o n t i n u o u s

Z - p a r t i c l e s . Examples of t h i s phenomenon a r e g i v e n i n Refs . 63 and 64.

Model f o r t he n r e d i c t i o n of t h e l a v e r seuuence

For a number of r e l a t i v e l y s i m p l e sys tems i t i s p o s s i b l e to p r e d i c t t he d i f f u s i o n p a t h

from the phase d i ag ram and k i n e t i c d a t a a c c o r d i n g to a model e x t e n s i v e l y d e s c r i b e d in Refs .

65-67 f o r s o - c a l l e d d i s p l a c e m e n t r e a c t i o n s of t h e t ype

A+BX~AX+B

a l t h o u g h more i n t r i c a t e r e a c t i o n s can a l s o be u n d e r s t o o d by u s i n g t h i s model .

R e f e r r i n g f o r d e t a i l s e s p e c i a l l y to Ref . [65] we w i l l J u s t summarise h e r e t he main i d e a s ,

u s i n g t h r e e r e a l s y s t e m s as examples . The most i m p o r t a n t s t a r t i n g p o i n t i s , t h a t no e l emen t

~ntrlnsically diffuses, i.e. jumps from one lattice site to another, up its own activity

gradient. The word "intrinsically" is very important here. An element does often

i n t e r d i f f u s e up i t s own a c t i v i t y g r a d i e n t , b u t t h a t t r a n s p o r t i s t h a n imposed upon t h i s

e l emen t by t he i n t r i n s i c d i f f u s i o n b e h a v i o u r of t h e o t h e r two: by t h e i r a tomic J , - ,ps from

one l a t t i c e s i t e to a n o t h e r t h e t h i r d e l e m e n t d r i f t s i n a d i r e c t i o n which i n d e e d may be up

i t s own a c t i v i t y g r a d i e n t . Th i s has m a t h e m a t i c a l l y been shown i n [ 4 2 ] .

The r e a d e r must be v e r y w e l l aware of t h e d i f f e r e n c e be tween t he a c t i v i t y g r a d i e n t and

t he c o n c e n t r a t i o n g r a d i e n t f o r an e l em en t i n a d i f f u s i o n c o u p l e . In a b i n a r y sy s t em, t h e s e

a lways have t h e same d i r e c t i o n . In a t e r n a r y s y s t e m , t h i s i s n o t t h e c a s e . In t he wel l -known

Darken e x p e r i m e n t [ 6 8 , 6 9 ] , f o r i n s t a n c e , u p - h i l l d i f f u s i o n of c a r b o n i s found from a

concentration point of view, but from an activity point of view carbon diffuses down its own

activity gradlent throughout the whole couple.

Let us take as a first example the reaction

Co + Cu20 • CoO + 2Cu

at 10OO'C. The relevant part of the phase diagram is shown in Fig. 3.2a, whereas the oxygen

activity from the Cu/Cu20 side towards the Co/CoO side is shown in Fig. 3.2.b. From the

phase stability diagram, the reaction proceeds because CoO is much more stable than Cu20.

FA F.J.J . van Loo

Diagram 3.2 .b i s cons t ructed f r ca a knowledge of the thermodynamic &O'-values for Co0 and

Cu20 , but may in a q u a l i t a t i v e way a lso be cons t ructed from the phase diagram by the slope

of the t i e - l i n e s [65].

~ , 8 "

=.. .

=o

~ o 0 Cu2 CoO 3 CuO

0

-6-

w

l -12

, CoO * Cu20

CoO ~" Cu

•] Co * Cu I ! " I |

Co .2 .4 .6 .8 (b)

Cu

•0. 8 Co CoO

~-10

"12 100 IJm I .J

(C)

1.2

i 8

. 4 \\\ \ \\,

{a)

\

nNi/nNi~nFe

-10"

.14 ̧

! | ~ NiO

f•"Fel-xO / 7 -(Fe,N0

! I F, . I .4 .6 .;

Ibl Ni

"E -10

Fe Fel.xO ~ ~ NiO (c)

Fig. 3.2 .

Fig. 3.3 .

a) Phase r e l a t i o n s and d i f f u s i o n path in Co-Cu-O system a t IO00"C. b) The oxygen pressure as a func t ion og the metal mole r a t i o . ~ e dots represent the oxygen pressure in equ i l ib r ium with the so l i d so l u t i ons Cu(Co) and Co(Cu). ¢) A schematic r e p r e s e n t a t i o n of the layer morpholo$y and equ i l ib r ium oxygen pressure in the couple Cu/Cu20 a f t e r a heat t reatment of 100 h a t IO00"C [65]. Phase r e l a t i o n s and d i f f u s i o n path (a) as well as the equ i l ib r ium oxygen pressure (b) in the Fe-Ni-O system a t 1000"C; ¢) represen ts the layer morphology and oxygen pressure in an Fe/NiO d i f f u s i o n couple a f t e r 100 h annea l ing a t 1000"C [65].

Multiphasc Diffusion 85

In F ig . 3 . 2 . c t h e a c t u a l l a y e r morphology i n a d i f f u s i o n c o u p l e i s sho~u. The a c t i v i t y of

oxygen i s c o n t i n u o u s l y r i s i n s f rme one s i d e t o t h e o t h e r w i t h o u t any e x t r m ~ m . The same

h o l d s f o r t h e a c t i v i t i e s of Co and Cu, which a r e n o t shown. Th i s means , t h a t f r o l a

thermodyuamic p o i n t of v iew t h i s morphology i s a l l o w e d , l e a d i n 8 to a d i f f t m i o n p a t h a s g iven

i n F i g . 3 . 2 . a . From a k i n e t i c p o i n t o f v iew no p rob lem e x i s t s , s i n c e t h e d i f f ~ m i o n of oxyKe~

t h r o u g h t h e r e s u l t i n g C u - l a y e r t o form Co0 i s v e r y f a s t [ 7 0 , 7 1 ] . The r a t e l i m i t i n g s t e p i s

t h e d i f f u s i o n of C o - i o n s t h r o u g h CoO, and i n d e e d q u a n t i t a t i v e p r e d i c t i o n s a b o u t t he t o t a l

l a y e r t h i c k n e s s t u r n o u t to a g r e e w i t h t h e e x p e r i m e n t s [ 6 5 , 7 0 ] .

The r e v e r s e p a t h , i . e . Co-Cu-CoO-Cu20, i s n o t f o r b i d d e n f o r thermody~mmic r e a s o n s , b u t

v e r y u n l i k e l y on k i n e t i c g r o u n d s . I t would mean t h a t oxygen i8 immobile and Co has to

d i f f u s e t h r o u g h t h e C u - l a y e r to form CoO and t h e s o l i d s o l u t i o n of Co i n Cu20, whereas Cu

i o n s have to move t h r o u g h Co0. These k i n e t i c s t e p s a r e much s l o w e r t h a n t h o s e i n t he l a y e r

s equence shown i n F i g . 3 . 2 . c .

In t he sys t em Fe-Ni-O, t h e phase d i ag ram i s a t ype r e l a t e d to Co-Cu-O i n t he s e n s e t h a t

t h e s l o p e s of a l l t i e l i n e s have t h e same s i g n , which l e a d s to a same type of p l o t f o r t he

oxygen a c t i v i t y a s a f u n c t i o n of t h e m e t a l r a t i o , a s shown i n F ig . 3 . 3 . a and b . A l a y e r

sequence F e / F e - o x i d e / N i / N i O i s t h e n p r e d i c t e d . The a c t u a l morphology of t he r e a c t i o n l a y e r ,

however , i s d i f f e r e n t i n t h e s e n s e , t h a t t h e i r o n o x i d e s a r e n o t formed i n a s t r a i g h t l a y e r

b u t p e n e t r a t e t h e formed Fe-Ni s o l i d s o l u t i o n i n a more o r l e s s f i n g e r - s h a p e d way. This

two-phased l a y e r was p r e d i c t e d and e x p e r i m e n t a l l y shown by Rapp and coworkers [70 ,72] and

c o n f i r m e d by ( 6 5 ] . The r e a s o n f o r t h i s morphology i s t h a t t h e d i f f u s i o n o f oxygen t h rough

t he Fe-Ni s o l i d s o l u t i o n i s now s l o w e r and r a t e - l i m i t i n g , compared to t h e f a s t d i f f u s i o n of

F e - i o n s t h r o u g h t h e o x i d e s . Then e a c h a c c i d e n t a l p e r t u r b a t i o n a t t h e F e - o x i d e / N i boundary

grows p r e f e r e n t i a l l y : t h e s t r a i g h t i n t e r f a c e i s n o t s t a b l e . The O r i g i n a l sequence

F e / F e - o x i d e s / N i / N i O can , however , s t i l l be r e c o g n i e e d a t t h e b o u n d a r i e s F e / F e - o x i d e and

Ni/NiO.

-12-

Cu2S * Ni3S 2

' - (Cu,Ni)

(b) ~ nN i /nNi ÷ nc u

Z c ! Cu .2 .4 .6 .8 Ni ~ u,Ni Cu2S

(a) ___....4~ nNil nNi ÷ nc u

l " " l s - i r ~ P ; I I " I

(c)

Fig. 3 .4 . Phase r e l a t i o n 8 and d i f f u s i o n p a t h ( a ) a s w e l l a s t h e e q u i l i b r i u m s u l f u r p r e s s u r e (b) i n t h e Cu-Ni-S sys tem a t 500"C; ( c ) r e p r e s e n t s t he l a y e r morphology and s u l f u r p r e s s u r e i n a Cu/Ni3S 2 d i f f u s i o n c o u p l e a f t e r 100 h a n n e a l i n g a t 500°C [65].

~6 F . J . J . van Loo

An e s s e n t i a l l y d i f f e r e n t phase diagram i s shown fo r the Cu-Ni-S system in Fig . 3 .~ .a and

b. Because of the changing s ign of the s lopes of the t i e l i n e s l eav ing from the m e t a l l i c

so l i d s o l u t i o n , a maximum in the s u l f u r a c t i v i t y i s found in the three-phase region

Cu2S-Ni3S2-Cu,Ni. Then s u l f u r i s thermodynamically not allowed to d i f f u s e i n t r i n s i c a l l y from

Ni3S 2 to the Cu-s ide , because then i t would d i f f u s e up i t s own a c t i v i t y g rad ien t . So a t

l e a s t a t the s t a r t of the process s u l f u r i s immobile, which leads to an i n i t i a l l ayer

sequence Cu/Cu,Ni/Cu2S/Ni3S 2 as indeed found [65].

The p r e d i c t i o n s from t h i s kind of cons ide ra t ions fo r reasonably uncomplicated systems

show a p e r f e c t agreement wi th the exper imental r e s u l t s , 16 of which are presented in

[65,66]. In more complicated systems, adapt ions have to be made such as shown for the

Fe-Nt-B and Fe-Co-B systems [73]. In s t emar i s ing , the morphology of the d i f f u s i o n zone in a

te rnary couple can be p red ic t ed by a schame such as shown in Fig . 3 .5 . Because complicated

systems have not ye t been s tud ied in t h i s r e spec t (with an except ion fo r the Ti-A1203 and

Ti-SiO 2 systems i n v e s t i g a t e d by B i l l e l [7&]), i t might be s a f e r to conclude tha t the model

can d e f i n i t e l y p r e d i c t which l aye r morphology i s ~ , leavin& open in some cases more

then one p o s s i b i l i t y which the model cannot decide [73]. Fur ther research i s t h i s f i e l d to

f ind a d d i t i o n a l c r i t e r i a i s p rogress ing . The expart~.-nts r e c e n t l y published by the group of

Chang [e .g : 75] on the i n t e r r a c i a l r e a c t i o n s between metals and Calliom Arsenide are

i n t e r e s t i n g in t h i s r e s p e c t , a l though in these r eac t i ons non-equ i l ib r ium s i t u a t i o n s a r i s e

a f t e r shor t anneal ing times or low temperatures . The phenomena repor ted for a Pt-CsAs

d i f f u s i o n couple a t 600"C [75] are complete ly in l i n e with the p r e d i c t i o n s by our model.

~X

.8-

+ =<

¢: .4

o ^ ,~.

sxl

I I .4 .6 .8 B

ax m I

" . 8 -

g +

-x

I .4,

0 A . .4 .6 .8

a) b)

c) o ~ . , , . ~ , d)

Fig. 3,5. Basic so rpho log ies of the r e a c t i o n zone fo r the dlsplacement r eac t ion A + BX~ B + AX in r e l a t i o n to two poss ib l e types of phase diagrams. In (a) the t i e l i n e s have the same s ign of s lopes , leading to morphologies shown in (c) depending on the k i n e t i c s in the system. In (b) the d i f f e r e n t s ign of the slopes of the t i e l i n e s permits only morphology (d) .


E x D e r i e ~ n t a l d e t e r m i n a t i o n of the r a t e - l i m i t i n e s tep

The r a t e - l i s i t i n g s t e p i n t h e d i f f u s i o n p r o c e s s can sometJJes be found i n a r a t h e r

e l e g a n t way by a p p l y i n g one of t h e r e a c t i o n p r o d u c t s a l r e a d y to one o f t h e s t a r t i n g

m a t e r i a l s b e f o r e making t he d i f f u s i o n c o u p l e [ 7 1 ] . For i n s t a n c e , i t was e x p e r i m e n t a l l y shown

t h a t the a p p l i c a t i o n of an e x t r a l a y e r of pure Cu on t he Cu20 end member i n a d i f f u s i o n

coup le w i t h Ni d i d n o t i n f l u e n c e t he t h i c k n e s s o f t h e newly formed Cu and N i 0 - r e a c t i o n

l a y e r s , p r o v i n g t h a t d i f f u s i o n of oxygen t h r o u g h Cu i s n o t t h e r a t e - l i m i t i n g s t e p . App ly ing ,

however , a N i O - l a y e r on t he N i - s u r f a c e b e f o r e making a d i f f u s i o n c o u p l e w i t h Cu20

d r a s t i c a l l y d e c r e a s e s t h e r e a c t i o n r a t e , p r o v i n g t h a t t he d i f f u s i o n of N i - i o n s t h r o u g h NiO

is the rate-limltlng step.

Anothe r way of f i n d i n g t h e r a t e - l i m i t i n g s t e p i s by i n v e s t i g a t i n g t h e s i d e s of t he

s t a r t i n g m a t e r i a l s which a r e n o t i n d i r e c t c o n t a c t w i t h t h e o t h e r end member. D i f f u s i o n

l a y e r s a r e o f t e n formed on t h e s e s i d e s by e v a p o r a t i o n of one o r more of t h e e l e m e n t s [65] .

This was found i n a Ni-Cu20 c o u p l e a n n e a l e d i n an e v a c u a t e d s i l i c a c a p s u l e i n which e x t r a

p i e c e s of Ni were p r e s e n t , as s c h e m a t i c a l l y shown i n F ig . 3 . 6 . From the f r e e s i d e s of t he

Cu20 end member oxygen e v a p o r a t e d and r e a c t e d w i t h t he e x c e s s of n i c k e l , f o rming a N iO- l aye r

which was e q u a l l y t h i c k as t h e one formed i n t he g e n u i n e s o l i d - s t a t e d i f f u s i o n c o u p l e . The

C u - l a y e r , however , formed a t t he o u t e r s i d e s of Cu20, was much t h i c k e r t han i n t h e

s o l i d - s t a t e d i f f u s i o n c o u p l e , t he r a t i o b / a a c t u a l l y b e i n g a b o u t 90. The r e a s o n i s c l e a r : in

the d i f f u s i o n c o u p l e t he C u - l a y e r c a n n o t grow f a s t e r t h a n a l l owed by t h e f o r m a t i o n of NiO,

which i s governed by t h e s low d i f f u s i o n of N i - i o n s . (From t h e r e a c t i o n e q u a t i o n and the

mola r volumes i t f o l l o w s t h a t t he t h i c k n e s s of t h e C u - l a y e r i s 1 .5 t i m e s t h e t h i c k n e s s of

the N i O - l a y e r ) . The t h i c k n e s s of t h e C u - l a y e r on t he f r e e s u r f a c e s of Cu20 i s dependen t on

the much f a s t e r d i f f u s i o n of oxygen atoms t h r o u g h Cu. The t h e o r e t i c a l l y c a l c u l a t e d r a t i o b / a

i s a b o u t 90 [ 7 0 ] , i n ag reemen t w i t h t h e o b s e r v e d v a l u e . In a Cu20-Fe d i f f u s i o n coup le t h i s

e f f e c t does n o t a p p e a r , s i n c e f o r b o t h t h e f r e e s u r f a c e s of Cu20 as w e l l as t h a t i n t he

d i f f u s i o n c o u p l e , t he d i f f u s i o n o f oxygen t h r o u g h Cu i s r a t e - l i m i t i n g .

A third possibility is the use of a diffusion couple in which an alloy of the metals is

used instead of a pure metal. This was demonstrated for the Ni-Cu-O system [63] by making

couples of the type NixCUl_x/CU20. For a value of x lower than about 0.~, Ni cannot be

supplied fast enough by diffusion in the end member alloy to form a continuous layer of NiO.

Then, if fact, internal oxidation occurs, and the diffusion of oxygen through the Cu-Ni

a l l o y i s r a t e - l i m i t i n g . The t o t a l l a y e r w i d t h ( a l l o y + N i O - p r e c i p i t a t e s ) i s now much l a r g e r

than i n t he c a s e where a c o n t i n u o u s N i O - l a y e r i s formed, p r o v i n g t h a t the d i f f u s i o n t h r o u g h

NiO was t he r a t e - l i m i t i n g s t e p i n a Ni-Cu20 c o u p l e .

~/ ..;-':t::i.......~.: ~:~li!i:411 ',,'--N i 0

m m m mm m m /

Fig . 3 . 6 . Schemat i c i l l u s t r a t l o n of t he l a y e r morphology i n a d i f f u s i o n c o u p l e Ni/Cu20 a t 1000*C, a n n e a l e d i n a c l o s e d s i l i c a c a p s u l e i n t h e p r e s e n c e of e x c e s s Ni .

88 F.J.J. van Loo

In t he same s y s t e m , a comparab le e f f e c t was found i f s m a l l amounts of c h l o r i d e i o n s were

p r e s e n t i n t h e C u 2 0 - h a l f of a Ni/Cu20 c o u p l e . These p r e v e n t e d t h e s i n t e r i n g of t h e

N i O - p a r t i c l e s to a c o n t i n u o u s l a y e r , l e a d i n g to randomly d i s p e r s e d NiO p a r t i c l e s and much

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

[64 ] .

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

Up to now we were d i s c u s s i n g t h e morphology of t he r e a c t i o n p r o d u c t s i n r e l a t i o n to t he

phase s t a b i l l t y d i ag rams m a i n l y from a thermodynamic p o i n t of v iew. In s l n g l e - p h a s e r e g i o n s ,

and e s p e c i a l l y i n s o l i d s o l u t i o n s formed from the s t a r t i n g m a t e r i a l s k i n e t i c f a c t o r s a r e

a l s o v e r y i m p o r t a n t a s shown i n [ 4 2 , 7 6 ] .

As a g e n e r a l t r e n d , i n t h e s i n g l e - p h a s e r e g i o n s of t h e end members, t h e d i f f u s i o n p a t h

s t a r t s i n t h e d i r e c t i o n of t h e f a s t e r d i f f u s i n g component i n t h e end member poor i n t h i s

component , and away from t h e f a s t e r d i f f u s i n g component i n t h e end member which i s r i c h in

t h i s component . I f t h e r a t i o o f t h e m o b i l i t t e s r ema ins t he same t h r o u g h o u t t h e whole sys t em,

t h i s r u l e , t o g e t h e r w i t h t h e c o n s t r a i n t s of t h e mass b a l a n c e , p e r m i t s a rough i d e a a b o u t the

c o u r s e of t h e d i f f u s i o n p a t h , which i s enough f o r many p r a c t i c a l p u r p o s e s . No te , however ,

t h a t t he d i f f u s i o n p a t h i n t h e c l a s s i c a l Darken e x p e r i m e n t ( F e - S i - C / F e - C ) c a n n o t be found

from t h i s k i n e t i c mode l , s i n c e t h i s p a t h f o l l o w s the d i r e c t i o n of d e c r e a s i n g ca rbon a c t i v i t y

and i s e x c l u s i v e l y d e t e r m i n e d by t he t he rmodynamics .

For i n s t a n c e , i n a phase d i ag ram such as F i g . 3 . 7 , where t h e e l e m e n t C i s t he f a s t e s t

d i f f u s i n g component i n t h e s i n g l e - p h a s e r e g i o n . I f we now c o n s i d e r a d i f f u s i o n c o u p l e of

pure A v e r s u s BC, t h e n t h e d i f f u s i o n p a t h can be e s t i m a t e d by t h e s e c o n s i d e r a t i o n s . The

p r e d i c t i o n of t h e e x a c t p o s i t i o n of p o i n t p, as w e l l as t h e e x a c t c u r v a t u r e of t h e d i f f u s i o n

p a t h , needs d e t a i l e d i n f o r m a t i o n a b o u t t he d i f f u s i v i t i e s of a l l e l e m e n t s and t h e i r mutua l

dependence . For t h e a u t h o r , t h i s c a l c u l a t i o n seems, g e n e r a l l y s p e a k i n g , a too f o r m i d a b l e

t a s k to u n d e r t a k e , and an e x p e r i m e n t a l d e t e r m i n a t i o n i s t h e o n l y way to o b t a i n an e x a c t

d i f f l t s i o n p a t h . B a c k h a u s - R i c o u l t [77,78], who pe r fo rmed b o t h e x p e r i m e n t a l and t h e o r e t i c a l

work on t h i s p rob lem came to t he same c o n c l u s i o n .

F ig . 3.7 .

C

BC

B

E s t i m a t e d d i f f u s i o n p a t h ( : : : . ) f o r a d i f f u s i o n coup le AIBC w i t h o n l y the knowledge t h a t t he e l e m e n t C i s t he most m o b i l e component . The t i e l i n e s a r e i n d i c a t e d as t h i n d o t t e d l i n e s .


I n f l u e n c e o f t he elm n a r t i a l o r e s s u r e on the d i f ~ u s i o n o a t h

As soon as Eases such as oxyEen and n i t r o E e n , b u t under some c i r c u m s t a n c e s a l s o s u l f u r or

phospho rus , t ake p a r t i n t h e r e a c t i o n , the gas at~eosphere d e c i d e s which r e a c t i o n s ~,ight

o c c u r . Take as an e .~mple t h e r e a c t i o n be tween Ni and A1203 ( s e e a l s o [ 7 8 , 6 6 ] ) . The phase

diagram in the N i - r i c h c o r n e r i s s c h e m a t i c a l l y shown in Fi E. 3 . 8 .

I f t he a n n e a l i n g o f a Ni/AI203 d i f f u s i o n coup le i s pe r fo rmed in an a tmosphere which i s

e x t r e m e l y low in oxygen p a r t i a l p r e s s u r e t he d i f f u m i o n pa th w i l l s t a r t from v i r t u a l l y the

pure N i - c o r n e r a l o n E p a t h 1. No new r e a c t i o n p r o d u c t i s formed and on ly a s o l i d s o l u t i o n of

A1 and O in Ni i s found . OxyEen p e n e t r a t e s much d eep e r i n t o n i c k e l than aluminum because of

t h e i r d i f f e r e n c e in d i f f u s i v i t y , bu t the a b s o l u t e v a l u e s of t he d i s s o l v e d amounts have to

obey t h e mass b a l a n c e , i . e . have to occu r in t he r a t i o 3 :2 .

I f , however , a n n e a l i n g t a k e s p l a c e a t an oxygen l e v e l which a l r e a d y c r e a t e s a s o l i d

s o l u t i o n of oxygen in pure Ni , the d i f f u s i o n p a t h s t a r t s a t a p o i n t d i f f e r e n t from the

c o r n e r p o s i t i o n , i . e . p a t h 2. Then, a c c o r d i n g to t he mass b a l a n c e , the p a t h might deve lop in

such a way t h a t s p i n e l i s formed. This i s what was p r e d i c t e d and e x p e r i m e n t a l l y found by

Backhaus -R tcou l t [78] .

This e f f e c t can p r o b a b l y i n f l u e n c e the morphology of the r e a c t i o n l a y e r dependen t on the

d i a m e t e r o f t he coup le h a l v e s . In Fi E. 3 .9 t he coup le pure Ni/A1203 i s a n n e a l e d in an oxyEen

a tmosphere equa l t o t h e e q u i l i b r i u m p r e s s u r e Ni/NiO. However, i t t a k e s t ime f o r oxygen to

d i f f u s e f a r enough to r e a c h p o i n t a in the c o u p l e . At p o i n t s b , on the o t h e r hand, a

s a t u r a t e d s o l i d s o l u t i o n o f oxygen in Ni i s r e a d i l y formed. T h e r e f o r e , one e x p e c t s to f i n d

e x p e r i m e n t a l l y s p i n e l f o r m a t i o n a t p o s i t i o n s b and on ly a s o l i d s o l u t i o n of oxygen and

aluminum in Ni a t p o i n t a . Whether or n o t t h i s i s found p r o b a b l y depends l a r g e l y on the

q u a l i t y of t he i n i t i a l c o n t a c t be tween Ni and A1203. I f t h e r e i s no t good c o n t a c t , oxygen

cou ld e a s i l y d i f f u s e a l o n e the i n t e r f a c e and form s p i n e l eve rywhere . The same ho lds f o r

r e a c t i o n s be tween e . E . Ni and Si3N 6 [79] . In our l a b o r a t o r y we a r e t r y i n s to s e t more

e x p e r i m e n t a l i n f o r m a t i o n on t h i s t r i p l e - p o i n t b e h a v i o u r .

Ni

0 C)

2 -phase -phase

; At

FiE. 3.8.

JPSSC 20:1-1;

D i f f u s i o n p a t h s f o r Ni/A1203 co u p l e s a t d i f f e r e n t oxygen p a r t i a l p r e s s u r e s [66,78].

90 F.J.J. van Loo

o2 (NqN O)

a

Ni

Fi&. 3.9. Predic ted d i f f e r e n t r e a c t i o n schemes in a Ni/A120 3 couple in the middle (a) and the t r i p l e po in t s (b) during anneal ing in an oxygen atmosphere equal to the pressure o£ the Ni/Ni0 equ i l ib r ium.

A r e l a t e d experiment fo r the Cu-Si-P system has a l ready been performed [80]. The react icm

between Cu3P emd Si appears to be l a r g e l y in f luenced by the phosphorus p a r t i a l p ressure , as

was shown by performing the r e a c t i o n a t 800 K in two ways, v i e . in a c losed s i l i c a v e s s e l or

in a dynemlc vacuum of 10 -9 a t=.

In the f i r s t case , the phases Cu3Sl end SiP were formed in the l aye r sequence p red ic ted

by our model according to the phase dlaKrem and the d l f f u s i v l t l e s as shown in Fig. 3.10 a

and b. Ins ide the capsule the p a r t i a l P4-pressure of SiP (about 0.03 atm.) was p resen t .

In the dynamic vacuum clrc~nnstances, the SiP phase i s not s t a b l e because of i t s high

p a r t i a l phosphorus pressure and the phase diagrem then looks l i k e Fig . 3.10 c. Accordin& to

the same model, the l aye r sequence as shown in Fig. 3.10 d has to be formed, whereby

phosporus evaporates a t the i n t e r f a c e Cu3P/Cu3SI. This indeed was found exper lmenta l ly [80].

p P~ gas / ~ dosed ~ open vac.

/ ~ system / / ~ system

• / / \ 10" atm.

SiP

Cu Si Cu ('/([ II ~\V/ \ Si Y Cu3Si Y Cu3Si

(b) Cu3P ~ Si (O)

P~ .gas

! P Cu3Si | Cu 3

J Fig. 3.10. D i f f e r e n t phase diagrams fo r the Cu-Si-P system and r e a c t i o n l aye r s in S i / C u ~

d i f f u s i o n couples when annealed in a c losed system (a and b) or in a dynamic vacuem of I0 - a ~ (c and d) [80].


The phosphorue gas c o u l d e s c a p e d e s p i t e a p p l y i n g an e x t e r n a l p r e s s u r e of 20 kg/cm 2 (2 NPa).

The coup le d i d n o t f a l l a p a r t . Pores were v i s i b l e a t t he i n t e r f a c e Cu3P/Cu3Si a t b i s h

t e m p e r a t u r e s where Cu3Si was r a t h e r d u c t i l e and , t h e r e f o r e , p l a s t i c a l l y de formed . At low

t a m p e r a t u r e s , no p l a s t i c d e f o r m a t i o n of Cu3Si o c c u r r e d , so t h a t p o r e s were p r e s e n t i n such

l a r g e d i m m s i o n e t h a t Cu3Si seemed to form i n i s o l a t e d columns be tween Cu3P and Si [80 ] .

I n f l m m e e of t h e s t o i e h i o m a t r v of the end members on the d i f f m a l o n Da~h

R e l a t e d t o t h e l a s t s u b j e c t i s t h e q u e s t i o n w h e t h e r s m a l l d e v i a t i o n s i n s t o i c h i c m e t r y

change a d i f f u ~ i o n p a t h . The answer depends on t h e sys tam a c t u a l l y a t hand , as s e e n by

t a k i n g two examples .

In t he c a s e of Ni /Cu20 t r e a t e d b e f o r e , t h e s t o i c h i c a e t r y of Cu20 i s n o t v e r y i m p o r t a n t .

I f Cu20 would be s l i g h t l y s u p e r s a t u r a t e d i n oxygen , CuO would be p r e s e n t f rom t h e b e g i n n i n g ,

b u t i t would be r educed to Cu20 by t h e f o r ~ a t i o n of NiO. I f t h i s i s o n l y a s m a l l ~ s o u n t , the

f u r t h e r r e d u c t i o n of Cu20 w i l l o c c u r and e v e n t u a l l y t h e l a y e r s equence Cu20/Cu/NiO/Ni w i l l

form. A l a r g e e x c e s s of oxygen would l e a d to a t h i c k N i O - l a y e r , d e l a y i n g t he f u r t h e r

r e d u c t i o n of Cu20 s i n c e t h e n e c e s s a r y t r a n s p o r t of Ni i o n s t h r o u g h NiO i s t oo s low.

I f Cu20 i s u n d e r s a t u r a t e d i n oxygen , i . e . i f some f r e e Cu i s p r e s e n t , no d i f f e r e n c e in

r e a c t i o n i s s een b e c a u s e t he p r e s e n c e of a Cu l a y e r does n o t i n f l u e n c e t he r e a c t i o n r a t e o r

l a y e r s e q u e n c e .

In o t h e r t y p e s of s y s t e m s , t he i n f l u e n c e of t he s t o i c h i o m e t r y m i g h t be l a r g e r . In the

sys tem T i - 8 i - C t he r e a c t i o n be tween Ti and SiC m i g h t be c o m p l e t e l y changed by t a k i n g SiC

w i t h e x c e s s Si o r C [ 8 1 ] . In t h e f i r s t c a s e , i t i s n o t p o s s i b l e to form t i t a n i u m c a r b i d e n e x t

to SiC, whereas i n t h e l a s t c a se t h i s i s q u i t e w e l l p o s s i b l e . Th i s e f f e c t p r o b a b l y a c c o u n t s

f o r t he l a r g e d i f f e r e n c e s g i v e n by v a r i o u s r e s e a r c h groups s t u d y i n g t h i s sys t em [ 8 2 - 8 ~ ] . In

ou r l a b o r a t o r y , we a r e c u r r e n t l y i n v e s t i g a t i n g t he sys t em e s p e c i a l l y i n t h i s r e s p e c t [81 ] .

The o c c u r r e n c e of a D e r l o d l c l a v e r f o r m a t i o n

In t h e i r r e c e n t book , K i r k a l d y and Young [85] g i v e a number o f examples in which a

p e r i o d i c p r e c i p i t a t i o n was found i n t e r n a r y and h i g h e r o r d e r s y s t m n s . They a r e r e l a t e d to

t he we l l -known L i e s e g a n g bands o b s e r v e d e . g . i n some o x i d a t i o n e x p e r i m e n t s .

In our l a b o r a t o r y O s l n a k l e t a l . [86 , 87] found examples i n t h e c o u p l e s Zn-Fe3Si and

Zn-Co2Si . Nehan and J a c k s o n [88] and S c h i e p e r s e t a l . [89] r e p o r t e d t h e same type of

s t r u c t u r e i n t he c a s e of t h e d i f f u s i o n coup le N i - S i C . R e c e n t l y , p e r i o d i c s t r u c t u r e s c l o s e l y

r e s e m b l i n g t h o s e i n Ni -S iC were r e p o r t e d f o r P t - S i C c o u p l e s [90 ] . I t i s i n t e r e s t i n g to

c o n s i d e r t h e s e s y s t em s i n a way c o m p l e t e l y d i f f e r e n t f rom K i r k a l d y ' s a p p r o a c h . In

F ig . 3 .11 a - c t he r e l e v a n t phase d iag rams a r e shown t o g e t h e r w i t h t h e d i f f u s i o n p a t h and

l a y e r sequence f o r s p e c i f i c d i f f u s i o n c o u p l e s , as p r e d i c t e d from our thermodynamic model .

The components Zn and Ni a r e t he o n l y d i f f u s i n g s p e c i e s ( a s shown by marker e x p e r i m e n t s ) ,

r e a c t i n g w i t h , r e s p e c t i v e l y , Fe , Co and Si to form t h e v a r i o u s i n t e r m e t a l l i c s FexZny, CoxZny

and NixS iy . The p r o d u c t s l e f t a r e t hen FeS i , CoSi and C, r e s p e c t i v e l y .

This s i t u a t i o n was i n d e e d found e x p e r i m e n t a l l y a f t e r s h o r t a n n e a l i n g t i m e s , whereby t he

phases FeS i , CoSi and C were formed as porous l a y e r s . A f t e r l o n g e r a n n e a l i n g t i m e s , however ,

t he d i f f u s i o n zones showed t h e FeS i , CoSi and C p h a s e s as porous l a y e r s , p r e c i p i t a t e d i n the

m a t r i c e s of FexZry, CoxZny and NixSiy a t r e g u l a r d i s t a n c e s and fo rming an i n t e r e s t i n g

p e r i o d i c s t r u c t u r e .


-3

-2

N

Zn Si

zn ,1 l 3s, (a)

Fhs~ ~ I j ~ 3 395"C

=u

Y', m ~ - - ' ~ - I ¥2Zn Si

I°[ ° Si C 2Si

(b)

NiSi;

NiSi

6 Y P_

(Ni) Ni C

0 l- ÷

Z

u~

=t Si C

N, j,rY N,,s,2r,jc[ s,c (c)

Fig. 3.11. Schematic i so thermal phase diagrams and the p red ic ted l aye r sequences in s p e c i f i c d i f f u s i o n couples in the systems Fe-Si-Zn (a ) , Co-SI-Zn (b) and Sl-C-Ni (c ) .


Our q u a l i t a t i v e e x p l a n a t i o n i s t he f o l l o w i n g , t a k i n g as an example the most i n t e n s i v e l y

s t u d i e d c a s e s FeHSi-Zn and SiC-Ni . Because of t he mechan i ca l s t r e s s e s b u i l t up a t the

i n t e r f a c e s be tween FeHSi and FeSi or SiC and C, r e s p e c t i v e l y , t h e s e phases can on ly grow to

a c e r t a i n t h i c k n e s s b e f o r e t h e y l o s e c o n t a c t w i t h t h e end member. Immedia te ly t h e r e a f t e r ,

the phases a r e formed a g a i n a t t h i s end meaber to t h e same l a y e r t h i c k n e s s , e t c . Then the

p e r i o d i c a l l y o c c u r r i n g phase s a lways s t a y in c o n t a c t w i t h one end member and a r e n e v e r i n

c o n t a c t w i t h the o t h e r one , which i s n o t t he c a s e i f a L iesegang mechanism i s o p e r a t i v e . The

s t r u c t u r e r e p o r t e d f o r P t - S i C [90] can be e x p l a i n e d i n the same way.

This i d e a was c o r r o b o r a t e d by e x p e r i m e n t s i n t he Fe -S i -Zn sys tem in which the Fe3Si

s t a r t i n g m a t e r i a l was v e r y t h i n and , t h e r e f o r e , c a p a b l e of d e f o r m a t i o n by mechanica l

s t r e s s e s . The i n i t i a l l a y e r sequence was s u s t a i n e d f o r much l o n g e r t i m e s , and the F e S i - l a y e r

grew to an a p p r e c i a b l e t h i c k n e s s b e f o r e the l a y e r l o s t c o n t a c t .

I f l e s s Si i s p r e s e n t in the s t a r t i n g m a t e r i a l , t he i n t e r d i f f u s i o n of Fe and Si i s no t

f a s t enough to form a l a y e r of FeSi a d j a c e n t to t he Fe-Zn compounds. The r e a c t i o n l a y e r

l o s e s i t s p e r i o d i c s t r u c t u r e , and a t low s i l i c o n c o n t e n t s , random p r e c i p i t a t e s of FeSi a r e

found in t he Fe-Zn i n t e r m e t a l l i c s .

For t he SiC-Ni s y s t e m s , the e x p e r i m e n t s w i t h t h i n SiC end members have n o t y e t been made

but a r e in p r o g r e s s . The c l o s e l y r e l a t e d SiC-Fe d i f f u s i o n co u p l e s show random p r e c i p i t a t e s

of g r a p h i t e in Fe3Si [89 ] , whereas in S iC-FexNi l_ x c o u p l e s , the bands and random

p r e c i p i t a t e s n i c e l y c o i n c i d e w i t h a l a r g e r o r s m a l l e r atom r a t i o of N i /Fe in t he r e a c t i o n

l a y e r [ 67 ,91 ] . This e f f e c t might be r e l a t e d to t h e d i f f e r e n t i n f l u e n c e of Fe and Ni on the

s i n t e r i n g b e h a v i o u r of g r a p h i t e .

In the coup le Co2Si/Zn [87] , a s t r a n g e e f f e c t was found; the phase CoSi formed in

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

d i r e c t i o n in the Co2Si c r y s t a l from which the r e a c t i o n p r o d u c t s o r i g i n a t e d .

I t would be w o r t h w h i l e to u n d e r s t a n d t h e s e phenomena in a more fundsmen ta l and

q u a n t i t a t i v e way, r a t h e r than in t he somewhat vague and q u a l i t a t i v e terms used h e r e .

D e t e r m i n a t i o n of a t e r n a r y i s o t h e r m a l c r o s s - s e c t i o n u s l n e d l f f u ~ i o n co u n l e s

S ince f o r each d i f f u s i o n coup le a un ique d i f f u s i o n p a t h i s f o l l o w e d , i n p r i n c i p l e , a

l a r g e number of d i f f u s i o n coup l e s must be s t u d i e d e x p e r i m e n t a l l y f o r a t e r n a r y i s o t h e r m .

Z P

A

C

T R

X Cl (a)

B P T Q

(b)

Fig . 3 .12 . D e t e r m i n a t i o n of an i s o t h e r m a l s e c t i o n of a phase diagram l i k e (a) from s e l e c t e d m u l t i p h a s e d i f f u s i o n c o u p l e s l i k e ( b ) .

9.~ F . J . J . van Leo

However, t h i s number can be d e c r e a s e d a p p r e c i a b l y by u s i n g p o l y p h a s e a l l o y s a s s t a r t i n g

m a t e r i a l s . The chance to h i t i n t e r f a c e s a t which t h r e e p h a s e s a r e i n e q u i l i b r i u m i s much

l a r g e r t h e n , b u t t h e s e t h r e e - p h a s e t r i a n g l e s have to be e x a c t l y known i n o r d e r to t m d e r s t a n d

t he phase d i ag ram. In F i g . 3 . 12 t h i s p r o c e d u r e i s shown s c h e m a t i c a l l y .

I f a d i f f u s i o n c o u p l e be tween t he a l l o y s P end Q s h o u l d show a morphology i n d i c a t e d i n

t h i s f i g u r e , t h e n f ram one c o u p l e two t h r e e - p h a s e t r i a n g l e s can be found . In a r e a 1,

m i c r o p r o b e measu remen t s r e v e a l t h e t h r e e - p h a s e t r i a n g l e T-Z-C, whereas from a r e a 2 t he

t r i a n g l e T-X-B can be found . By making sandwich c o u p l e s l i k e P/Q/R even more i n f o r m a t i o n can

be g a i n e d . In t h i s way, a few c o u p l e s can g i v e a l l t h e n e c e s s a r y i n f o r m a t i o n .

E x t e n s l o n to m m t e r n a r v Ivet~uns

The g r a p h i c a l d i f f i c u l t i e s i n p l o t t i n g a d i f f u s i o n p a t h i n a q u a t e r n a r y phase d iagram i n

a c l e a r way a r e somet imes f r u s t r a t i n g . In some s i m p l e sy s t ems such as t h e F e - N i - S i - C sys tem

men t ioned b e f o r e [67 ,91 ] a c l e a r r e p r e s e n t a t i o n i s p o s s i b l e . By t a k i n g t he F e - N i - S i t r i a n g l e

as a b a s e and u s i n g NC/NFe+NNi+NSi a s a v e r t i c a l a x i s , t he sys t em can be p r e s e n t e d as shown

i n F ig . 3 .13 a . A l l p h a s e s a t t h e S i - s i d e of t h e b r o k e n l i n e i n t h e b a s e t r i a n g l e a r e i n

e q u i l i b r i u m w i t h SiC, t he o t h e r c o m p o s i t i o n s a r e i n e q u i l i b r i o m w i t h g r a p h i t e . As a

consequence o f t h e thermodynamic mode l , o n l y t h e s e l a t t e r c o m p o s i t i o n s can be formed i n

d i f f u s i o n c o u p l e s o f t h e t ype S i C / FexN i l _ x . For a number of t h e s e c o m p o s i t i o n s , t h e

p r o j e c t i o n s of t h e e x p e r i m e n t a l d i f f u s i o n p a t h s on t he b a s e t r i a n g l e a r e shown i n

F ig . 3 .13 b [ 9 1 ] .

I f q u a t e r n a r y p h a s e s a r e fo rmed , t h e v i s u a l i z a t i o n i s more d i f f i c u l t . Ye t , t h i s way of

p r e s e n t i n g d i f f u s i o n p a t h s i s b e t t e r t h a n u s i n g t h e u s u a l p y r a m i d - t y p e of a q u a t e r n a r y phase

d i a g r a m .

4 . CONCLUSIONS

The r e a d e r will have n o t i c e d t h a t t h i s p a p e r , e x p e c l a l l y t h e p a r t on t e r n a r y s y s t e m s , was

q u i t e q u a l i t a t i v e i n c h a r a c t e r . I t i s i n d e e d i n t e n d e d to p r e s e n t a f e e l i n g f o r t h e p r o c e s s

of m u l t l p h a a e d i f f u s i o n , f o r t h e p o s s i b i l i t i e s of p r e d i c t i n g which r e a c t i o n s migh t o c c u r ,

and f o r t h e e x p e r i m e n t a l means to s e t u s e f u l i n f o r m a t i o n .

The a u t h o r i s f u l l y aware of t h e f a c t t h a t s e v e r a l good books have been w r i t t e n on t h i s

s u b j e c t , l i k e e . g . [~ , 8 5 ] . I t i s , however , h i s e x p e r i e n c e w i t h s t u d e n t s and s c i e n t i s t s

a f t e r a number of y e a r s t h a t m a t h e ~ t i c a l l y c o r r e c t d e s c r i p t i o n s o f , e s p e c i a l l y t e r n a r y ,

d i f f u s i o n p r o c e s s e s a r e too i n t r i c a t e to be r e a l l y u s e d . I f one wan t s to know e x a c t l y what

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

c o n c e n t r a t i o n and t e m p e r a t u r e - d e p e n d e n c e , have to be known. I t i s t h e a u t h o r ' s f e e l i n g t h a t

t h i s g i a n t t a s k to g a t h e r t h e s e d a t a , w i l l n o t i f e v e r be comple t ed soon . T h e r e f o r e , a

q u a l i t a t i v e f e e l i n g a b o u t m u l t i p h a s e d i f f u s i o n r ema ins v e r y i m p o r t a n t f o r t he coming y e a r s ,

and t he a u t h o r hopes to have c o n t r i b u t e d to t h a t aim by t h i s work.


c. C C

Ni

Fe Ni

b)

Fig. 3.13. a) The isothermal quaternary phase diagram Fe-Ni-Si-C, exper imenta l ly determined at 850"c [91] b) d i f f u s i o n patha projected on the Fe-Ni-Si sect ion of the couples SiC/FesoNi20, SiC/Fe6oNi~o , SiC/Fe40Nt60 and SiC/Fe20Ni80 [67,91].

96 F.J.J. van Loo

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LIST OF SYMBOLS

a = u n r e a c t e d r a d i u s o r h a l f t he u n r e a c t e d p l a t e t h i c k n e s s in m, Eq. 24

a i = activity of component i=YiN i

C i = concentration of component i in moles/m 3

Ci = i n i t i a l c o n c e n t r a t i o n of component i on the l e f t - h a n d s i d e of the

d i f f u s i o n coup le

C~ = i n i t i a l c o n c e n t r a t i o n of component i on the r i g h t - h a n d s i d e o f the 1

d i f f u s i o n coup le

= i n t e r d i f f u s i o n c o e f f i c i e n t ( a l s o c a l l e d chemica l d i f f u s i o n c o e f f i c i e n t ) in m2/s , Eq. 2

D i = i n t r i n s i c d i f f u s i o n c o e f f i c i e n t of component i in m2/s , Eq. 11 T D i = t r a c e r d i f f u s i o n c o e f f i c i e n t of component i in m2/s , Eq. 13

= ave rage i n t e r d i f f u s i o n c o e f f i c i e n t in m2/s , Eq. 17 av

D i n , " i n t e g r a t e d i n t e r d i f f u s i o n c o e f f i c i e n t in m2/s , Eq. 17

d = t o t a l l a y e r t h i c k n e s s in m

d i = l a y e r t h i c k n e s s a f t e r a t r a n s i e n t p e r i o d t i , Eq. 18

dp = l a y e r t h i c k n e s s on a p l a t e du r ing the p a r a b o l i c growth p e r i o d , Eqs. 18 and 25

dc, dp and d s = l a y e r t h i c k n e s s e s in c y l i n d r i c , p l a n a r and s p h e r i c a l

diffusion couples during diffusion-llmlted growth, Eq. 25

J i = i n t e r d i f f u s i o n f l u x of component i in the Matano frame of r e f e r e n c e in mole l s /m2s ,

Eq. l

J i = i n t r i n s i c d i f f u s i o n f l u x of component i in the K i r k e n d a l l frame of r e f e r e n c e in

moles /m2s, Eq. 4


kp =

k' i

P N i =

R =

r i =

r 0

T =

t =

t i =

V m =

vKIM=

X ~e

X O

x K =

Y =

Z =

CL

7i =

p a r a b o l i c growth c o n s t a n t i n m2/s , Eq. 18

a p p a r e n t v a l u e of k from a d / t ~ p l o t , Eq. 19 P

mole f r a c t i o n of component i . For s u p e r s c r i p t - o r +, s ee C i

homogenei ty w i d t h of a phase

gas c o n s t a n t = 8 ,314 J / m o l e K

d i m e n s i o n l e s s f a c t o r o r i g i n a t i n g from the vacancy f l u x e f f e c t , Eq. 13

r a d i u s o f c y l i n d e r or s p h e r e , o r h a l f the p l a t e t h i c k n e s s in m, Eq. 24

t e m p e r a t u r e in Ke lv in u n l e s s s t a t e d o t h e r w i s e

diffusion time in seconds

duration of the transient period before the parabolic growth sets in, in s, Eq. 18

molar volume in m3/mole of atoms

partial molar volume of component i in m3/mole of atoms

velocity of local inert markers relative to the Matano frame of reference in m/s,

Eq. 5

c o o r d i n a t e in the d i f f u s i o n d i r e c t i o n in m

c o o r d i n a t e of t he o r i g i n of t he Natano frame of r e f e r e n c e

c o o r d i n a t e of the K i r k e n d a l l p l ane

x K - x 0 in m, Eq. 5

concentration unit for component i = (Ni-Ni-)/(Ni+-Ni-), Eq. 3

ratio between the volume of one mole ABy and one mole A, Eq. 25

conversion factor, Eq. 24

activity coefficient of component i

C i * o r Y*) r e f e r s to a d e f i n i t e v a l u e of the q u a n t i t y i n q u e s t i o n . An a s t e r i s k (as i n

The s u p e r s c r i p t N as in Eq. 16 d e n o t e s q u a n t i t i e s i n the n u m b e r - f i x e d frame of r e f e r e n c e .

multiphase diffusion in binary and ternary solid-state systems · multiphase diffusion in binary...

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