A MODIFICATION OF DARKEN'S EQUATION FOR THE INTERDIFFUSION COEFFICIENT IN P-TYPE OXIDE SOLID SOLUTIONS

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A MODIFICATION OF DARKEN’S EQUATION FOR THE INTERDIFFUSION COEFFICIENT IN P-TYPE

OXIDE SOLID SOLUTIONS

F. Gesmundo, F. Viani

To cite this version:

F. Gesmundo, F. Viani. A MODIFICATION OF DARKEN’S EQUATION FOR THE INTERDIF-

FUSION COEFFICIENT IN P-TYPE OXIDE SOLID SOLUTIONS. Journal de Physique Colloques,

1986, 47 (C1), pp.C1-543-C1-548. �10.1051/jphyscol:1986182�. �jpa-00225613�

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A MODIFICATION OF DARKEN'S EQUATION FOR THE INTERDIFFUSION COEFFICIENT IN P-TYPE OXIDE SOLID SOLUTIONS

F. GESMUNDO and F . VIANI*

Istituto di Chimica Fisica Applicata dei Materiali, C.N.R., Lungobisagno Istria 34, I-16141 Genova, Italy

' ~ s t i t u t o di Chimica, Facoltd di Ingegneria, Universitd di Genova, Fiera del Mare, Pad. D , I-16129 Genova; Italy

Resum@

-

Une forme m o d i f i e e de 1 ' 6 q u a t i o n de Darken pour l e c o e f f i c i e n t d ' i n t e r d i f f u s i o n e s t obtenue pour l e c a s de s o l u t i o n s s o l i d e s d'oxydes de t y p e (A,B)O en considerant 1 ' e f f e t de 1 a d i f f e r e n c e e n t r e l e s c o e f f i c i e n t s de d i f f u s i o n des deux i o n s sur l a c o n c e n t r a t i o n des defauts r e t i c u l a i r e s pendant 1 'equi 1 i b r a t i on.

A b s t r a c t - A m o d i f i e d form o f Darken's equation f o r 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 s obtained f o r t h e case o f s o l i d s o l u t i o n oxides o f t h e

(A,B)O t y p e by considering t h e e f f e c t o f t h e d i f f e r e n c e between t h e d i f - f u s i o n c o e f f i c i e n t s o f t h e two i o n s on t h e c o n c e n t r a t i o n o f t h e l a t t i c e d e f e c t s d u r i n g e q u i l i b r a t i o n .

I

-

INTRODUCTION

I n t e r d i f f u s i o n experiments a t constant temperature and oxygen a c t i v i t y between two oxides which are completely s o l u b l e can be used t o o b t a i n t h e i n t e r d i f f u s i o n coef- f i c i e n t o f t h e system, which i s a parameter d e s c r i b i n g t h e r a t e o f mixing o f t h e two components. 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 i n b i n a r y m e t a l l i c systems i s r e l a t e d t o o t h e r phenomenological parameters measured independently, i . e . t h e 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 o f t h e two components, through an equation o r i g i n a l l y d e r i v e d by Darken /I/. A p p l i c a t i o n o f t h i s equation t o c a l c u l a t e 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 f o r b i n a r y - o x i d e s o l i d s o l u t i o n s o f t h e (A,B)O t y p e has been shown t o present soae pro- blems /2-4/. I n f a c t d i f f u s i o n i n oxide systems i s d i f f e r e n t from d i f f u s i o n i n me- t a l s due t o requirement t h a t e l e c t r i c a l n e u t r a l i t y i s maintained and due t o t h e i n - f l u e n c e o f a f u r t h e r parameter, t h e value o f t h e oxygen a c t i v i t y . I n f a c t , unless t h e 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 o f t h e two i o n s happen t o be t h e same f o r equal values o f t h e oxide composition and oxygen a c t i v i t y , t h e two i o n s w i l l d i f f u s e i n opposite d i r e c t i o n s a t d i f f e r e n t r a t e s , producing a d e v i a t i o n o f t h e c o n c e n t r a t i o n o f vacancies from t h e corresponding e q u i l i b r i u m values i n t h e two sides o f t h e d i f - f u s i o n couple. This i n t u r n produces l o c a l changes o f t h e oxygen a c t i v i t y which af- f e c t t h e f l u x e s o f t h e two ions, tending t o a v o i d t h e p r o d u c t i o n o f l a r g e d e v i a t i o n s frorn t h e e q u i l i b r i u m i n s i d e t h e sample. The f o r m a t i o n o f oxygen a c t i v i t y g r a d i e n t s connected w i t h t h e d i f f e r e n t d i f f u s i v i t i e s o f t h e c a t i o n s i n oxide s o l i d s o l u t i o n s has been b o t h proposed on t h e o r e t i c a l grounds and confirmed e x p e r i m e n t a l l y /2-4/.

The purpose o f t h e present work i s t o take i n t o account t h e development o f oxygen

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986182

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c1-544 JOURNAL DE PHYSIQUE

a c t i v i t y g r a d i e n t s i n s i d e t h e d i f f u s i o n couple i n an i n t e r d i f f u s i o n experiment i n an (A,B)O s o l i d s o l u t i o n t o o b t a i n a m o d i f i e d form o f Darken's equation.

To r e s t r i c t t h e treatment, t h e oxides are assumed t o have t h e same k i n d o f e l e c t r i - c a l c o n d u c t i v i t y (p-type), t o c o n t a i n o n l y metal vacancies as l a t t i c e d e f e c t s and t o form a complete s e r i e s o f s o l i d s o l u t i o n s e x i b i t i n g i d e a l behavior: i n a d d i t i o n , i t i s assumed t h a t oxygen d i f f u s i v i t y i s n e q l i g i b l e w i t h respect t o t h a t o f t h e two me- t a l components.Al1 t h e previous c o n d i t i o n s apply t o t h e oxides considered here f o r

numerical a p p l i c a t i o n s . I 1

-

^THEORY

The d i f f u s i o n f l u x o f t h e two types o f ions A and B i n an (A,B)O s o l i d s o l u t i o n under t h e most general case i n v o l v i n g t h e presence o f g r a d i e n t s o f oxygen a c t i v i t y and oxide composition has been given by C. Wagner /5/, i n connection w i t h t h e a n a l y s i s o f t h e p a r a b o l i c growth o f an (A,B)O oxide on a b i n a r y A-B a l l o y , i n t h e f o r m

and

.

a l n a

BO ) a

- -

aE

I

aE 0 ax ax

where C i s t h e mole f r a c t i o n o f SO i n t h e mixed oxide, D and D are t h e d i f f u s i o n c o e f f i c i e n t s o f t h e two i o n s i n t h e oxide s o l u t i o n , a Aa an8 a are t h e a c t i v i - t i e s o f AO, 80 and oxygen i n t h e oxide, respectively,li%d

F0 i j

t h e o v e r a l l metal 0 c o n c e n t r a t i o n i n (A,B)O expressed as moles per u n i t volume rcm

1.

The d i f f u s i o n c o e f f i c i e n t s appearing i n Eqs. ( 1 ) and ( 2 ) are considered t o be t h e 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 of t h e two ions, i n agreement w i t h t h e general expression o f 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 /S/, w h i l e t h e fluxes are given w i t h r e s p e c t t o a l o c a l l a t t i c e plane. Assuming t h a t t h e oxide s o l u t i o n i s thermodynamically i d e a l ( i .e. aAO (1

-

E) and ago = E), Eqs. (1) and ( 2 ) reduce t o

I t should be r e c a l l e d t h a t D and D i n a mixed oxide w i l l depend b o t h on t h e oxygen

A B

a c t i v i t y and on t h e oxide composition / 7 / m a i n l y because they are p r o p o r t i o n a l t o t h e o v e r a l l c o n c e n t r a t i o n o f metal vacancies i n p-type oxides.

During an i n t e r d i f f u s i o n experiment, two samples o f s o l i d s o l u t i o n w i t h d i f f e r e n t composition ( b u t each homogeneous per se) and w i t h a constant common value o f t h e oxygen a c t i v i t y are brought together under a constant oxygen a c t i v i t y i n t h e gas phasein e q u i l i b r i u m w i t h t h e a i n t h e oxides. D i f f u s i o n between t h e two r e g i o n s takes p l a c e due t o t h e d i f f e r e n c e i n oxide composition. I n p r i n c i p l e , t h e two f l u x e s 3 are given by

and

so t h a t t h e two f l u x e s should d i f f e r i f D # DB. However t h i s c o n d i t i o n leads t o a s i t u a t i o n o f non equi 1 i b r i u m concerning t f i e c o n c e n t r a t i o n o f d e f e c t s i n t h e various regions o f t h e sample. The extreme case f o r t h i s i s when t h e f a s t e r d i f f u s i n g i o n

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regions o f concentrations o f A higher than t h e average f i n a l value (

f

) more r a - p i d l y than t h e slower d i f f u s i n g species B w i l l d i f f u s e i n . As a r e s u l t t h e concen- t r a t i o n o f metal vacancies w i l l r i s e i n t h e A-rich regions, where i t should i n s t e a d decrease as a r e s u l t o f t h e increase o f t h e c o n c e n t r a t i o n o f 5. The opposite s i t u a - t i o n w i l l occur i n regions c o n t a i n i n g a c o n c e n t r a t i o n o f A lower than t h e average, where t h e c o n c e n t r a t i o n o f vacancies w i l l decrease as a r e s u l t o f t h e d i f f u s i o n p r o - cess. I n t h i s way a d e v i a t i o n from t h e c o n d i t i o n o f e q u i l i b r i u m f o r t h e concentra- t i o n o f vacancies w i l l be produced on b o t h sides o f t h e sample, I n p r i n c i p l e , vacan- c i e s c o u l d be destroyed

in

r e g i o n s where t h e y are supersatured, and c o u l d be created where t h e i r c o n c e n t r a t i o n i s below e q u i l i b r i u m , b u t these processes are n o t impor- t a n t i n oxides /3,4/. The l o c a l change i n t h e c o n c e n t r a t i o n o f vacancies can be i n - t e r p r e t e d as a corresponding change i n t h e oxygen a c t i v i t y . I n f a c t , an oxygen a c t i - v i t y g r a d i e n t has been shown t o be produced i n d i f f u s i o n couples i n systems o f t h i s k i n d /2-4/, and i t has been proposed t h a t t h i s i s such t o make 3 % -JB /2-4/. Use o f t h i s c o n d i t i o n i n c o n j u n c t i o n w i t h t h e general expression f o r t h e two f l u x e s g i - A ven above y i e l d s

from which, a f t e r rearrangement, t h e r a t i o between t h e g r a d i e n t s o f I n a. and E i s obtained i n t h e form

a l n ag _{a x}

12

⁼ _{D A ( l}D - 0 ^B

_-

_{E )}^A

₊

_DBC ⁼ ^a Under t h i s c o n d i t i o n , t h e two f l u x e s then become

JA = cM DA

[I

⁺^{( 1}

-

E l a ] (aE/ax) and

J B =

-

c M D B [ 1

-

a ~ ] ( a ~ 8 x ) ( 6 )

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 f o r t h e system i s d e f i n e d by t h e equation /8/

7 . ^JA^E,

-

J B ( l - E l

D ( I ) =

-

. ( aE/ax)

which, upon i n t r o d u c t ~ o n o f Eqs. ( 5 ) and (6), becomes

D ( I ) = DA5

+

D B ( l - Z , ) + a E , ( l - 5 ) ( D A

-

DB) ( 7 The normal expression o f 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 , n e g l e c t i n g t h e f a c t o r S a r i s i n g from t h e vacancy f l o w e f f e c t , which i s u s u a l l y considered very c l o s e t o one /2-4,6/, i s

a l n yi D(I) = [ D ~ c + ~ ~

-

(

E )

1

]

(1 +

a l n xi

1

where t h e f a c t o r

= 1

+

( a l n y . / a l n x . )

1 1

( w i t h yi= a c t i v i t y c o e f f i c i e n t o f t h e component i (A o r B) and x. i t s mole f r a c t i o n ) i s t h e s o - c a l l e d thermodynamic f a c t o r . I n t h e case o f an i d e a l s o l u t i o n t h i s f a c t o r 1 i s considered equal t o one n o t o n l y f o r a b i n a r y system A-B, b u t a l s o f o r a t e r n a r y system such as a s o l i d s o l u t i o n between A0 and BO /2-4/. The expression obtained above d i f f e r s from t h e usual form o f Darken's equation, i n which 6 i s considered e- qual t o one, by t h e presence o f t h e c o r r e c t i o n term

B = a E ( 1 - E ) ( D A - DB).

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c1-546 JOURNAL DE PHYSIQUE

It i s seen immediately t h a t t h i s term i s equal t o zero when D = DB, b u t t h a t i t i s always negative when D # D thus making D(1) c a l c u l a t e d according t o Eq. A ( 7 ) a l -

A B Y .

ways smaller than t h e value given by Darken's equation.

It i s a l s o i n t e r e s t i n g t o p o i n t o u t t h a t , according t o t h e expression o f a r e p o r t e d above, t h e g r a d i e n t o f oxygen a c t i v i t y w i l l have t h e same s i g n as t h a t o f

c

i f DB>

D

,

b u t opposite t o i t i f DB<D For p-type oxides c o n t a i n i n g metal vacancies as t h e p r e v a i l i n g d e f e c t s t h i s imp'?;es t h a t i n t h e f i r s t i n s t a n c e t h e r e w i l l be a n e t f l u x o f vacancies from a r e g i o n o f h i g h a values ( h i g h concentrations o f

B)

t o t h a t o f low a0 values (small concentrations o f B), and a n e t metal f l u x i n t h e opposite 0 d i r e c t i o n . T h i s w i l l t h u s produce a decrease o f t h e f l u x o f

B

and an increase o f t h a t o f A u n t i l t h e y are almost equal. The opposite s i t u a t i o n occurs i f D < D since i n t h i s case t h e g r a d i e n t o f a w i l l increase t h e f l u x o f B and decrease !hat o f A. A It i s a l s o p o s s i b l e t o o b t a i n

eq.

^{( 7 )}for D(1) by s t a r t i n g from t h e d e f i n i t i o n o f 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 b u t t a k i n g i n t o account t h e c o r r e c t values f o r t h e thermodynamic f a c t o r s f o r t h e two metal ions i n t h e s o l i d s o l u t i o n . I n f a c t , as shown elsewhere /9/, t h e thermodynamic f a c t o r s f o r t h e metal i o n s A and B i n an o x i - de s o l i d s o l u t i o n are n o t always equal t o one, and i n general they d i f f e r from each other, t h e actual values depending on t h e oxygen a c t i v i t y , on t h e oxide composition and a l s o on t h e r a t i o between t h e g r a d i e n t s o f t h e s e two v a r i a b l e s . This r e s u l t shows c l e a r l y t h a t t h e e r r o r connected-with t h e use o f Darken's equation f o r t h e estimate o f D ( 1 ) from t h e 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 o f A and B depends on t h e approxima- t i o n o f c o n s i d e r i n g t h e thermodynamic f a c t o r s o f t h e two i o n s equal t o each other and equal t o one. I n f a c t , as shown elsewhere /9/, one obtains 4 = 19 =1 o n l y when d i f f u s i o n i n t h e mixed oxide occurs as a r e s u l t o f t h e presence 04 g r a d i e n t s o f

B

oxide composition under constant oxygen a c t i v i t y . I n t h e i n t e r d i f f u s i o n experiments however t h i s simp1 e c o n d i t i o n does n o t apply, as examined above, so t h a t t h e two thermodynamic f a c t o r s are no longer equal t o one.

I 1 1

-

NUMERICAL APPLICATIONS

The equations d e r i v e d i n t h e previous s e c t i o n are a p p l i e d here t o a s p e c i f i c system f o r which a l l t h e r e l e v a n t data have been measured, i . e . s o l i d s o l u t i o n s between N i O and COO. I n t h i s case t h e l i m i t i n g pure oxides are p-type semiconductors c o n t a i n i n g metal vacancies

/ l o / ,

w h i l e t h e behavior o f t h e two oxide components i n t h e s o l i d s o l u t i o n i s p r a c t i c a l l y i d e a l /11/.

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 i n NiO-COO s o l i d s o l u t i o n s has been measured a t d i f - f e r e n t temperatures /11/, b u t simultaneous d i r e c t measurements o f t h e 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 t h e two c a t i o n s Dvi and DCo have been c a r r i e d o u t o n l y a t 1300' and 1445°C /2/; t h e r e f o r e comparison between c a l c u l a t e d and experimental values can be c a r r i e d o u t o n l y a t these temperatures.

The 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 b f Ni and Co a t 1300°C a t t h e o x y g e n a c t i v i t y o f a i r have been measured f o r s e l e c t e d values o f t h e mole f r a c t i o n o f Co i n t h e mixed oxide. I n t h e f o l l o w i n g , N i i s considered as A and Co as t h e

B

ion. The 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 can be expressed as f u n c t i o n s o f i n t h e form /2/

DNi =

Do.

exp(a c + blc 2 ) and 2

N ¹ 1 DCo = D t o exp(a2c + b 2 c

where D o and Do are t h e values o f D and D a t

c

⁼0.

Values

1)

a

,

bCo a2 and b given by I d e n e t

g?.

/2/ do n o t represent t h e actual da- t a v e r y w e l l ; i t ' i s found ghat b e t t e r agreement w i t h t h e experimental d a t a i s o b t a i - ned by using i n s t e a d s l i g h t l y m o d i f i e d values, which are

a = 5.653; b, = -1.444; a = 5.9581; b = -1.6195

Use o f these expressions f o r D 1 and DCo as funcgions o f allows one t o evaluate a

N i

and then D ( I ) as f u n c t i o n s o f

E.

The r e s u l t s o f t h e c a l c u l a t i o n are r e p o r t e d i n F i g . 1 along w i t h t h e curve o f D(1) corresponding t o Darken's equation and w i t h t h e expe-

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curve o f D(1) c a l c u l a t e d according t o Eq. (7) i s c l e a r l y lower than t h a t corresponding t o Darken's curve, t h e d i f f e r e n c e being l a r g e s t around = 0.5, because t h e c o r r e c - t i o n term o f Ea. ( 7 ) reduces t o zero a t

5

= 0 and

5

= 1. The experimental data of D ( I ) are t o o low f o r

5

= 0.51 and p a r t i c u l a r l y f o r

c

⁼0.76 because i n t h e l a s t i n - stance t h e data f o r

5

= 0.76 f a l l below t h e smallest D

,

i . e . DNi. However, t h e a- greement w i t h t h e experimental value o f D(1) a t 1307°C i s r a t h e r good. Even i f t h e d i f f e r e n c e between t h e two curves f o r D(1) i s l i m i t e d , t h e change produced by t h e c o r r e c t i o n term i s i n t h e c o r r e c t d i r e c t i o n . I n a d d i t i o n , i t can be stressed t h a t t h e importance o f t h e c o r r e c t i o n i s r e l a t e d t o t h e d i f f e r e n c e between t h e two d i f f u - s i o n c o e f f i c i e n t s , being p r o p o r t i o n a l t o (DB

-

tlAl2. Thus t h e e f f e c t o f t h i s term i s r e l a t i v e l y low i n t h i s case where DB/DA i s r a t h e r small (from 2.1 t o 2.42), b u t i t should be more important when DB/DA i s very d i f f e r e n t from t h e u n i t y .

The expressions given f o r t h e two 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 a t 1445°C are s u f - f i c i e n t l y c o r r e c t . F i g . 1 shows t h e curves f o r D(1) according t o Darken's equation and according t o Eq. ( 7 ) a l s o a t t h i s temperature. I n t h i s case t h e d i f f e r e n c e b e t - ween t h e two curves o f D(1) i s smaller than t h a t a t 1300°C because t h e DB/DA r a t i o i s a l s o smaller. A t t h i s temperature t h e agreement between t h e experimental and c a l - c u l a t e d d a t a f o r D ( I 1 i s b e t t e r than t h a t a t 1300°C.

F i g . 1

-

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 D ( I ) a t 1300 and 1445OC i n a i r versus

5.

Curves a: c a l c u l a t e d from Eq. ( 8 ) ; curves b: c a l c u l a t e d from Eq. (71; 0 , A and A : experimental data from S t i g l i c h e t a l . /11/ a t 1436, 1307 and 129g°C, r e s p e c t i v e l y . REFERENCES

/1/ Darken, L., Trans. AIME 175 (1948) 184.

/2/ Chen, W.K. and ~ e t e r s o n , X ~ . , J. Phys. Chem. S o l i d s

2

(1973) 1093.

/3/ Yurek,

G.J.

and Schmalzried, H., Ber. Bunsenges. Physik. Chem. 78 (1974) 1379.

/4/ Yurek, G.J. and Schmalzried, H., Ber. Bunsenges. Physik. Chem. (1975) 255.

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