CALCULATION OF THE PARTIAL STRUCTURE FACTORS IN BINARY LIQUID ALLOYS BY A THERMODYNAMICAL APPROACH

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CALCULATION OF THE PARTIAL STRUCTURE

FACTORS IN BINARY LIQUID ALLOYS BY A

THERMODYNAMICAL APPROACH

M. Favre-Bonte, J. Joud, P. Hicter, P. Desre

To cite this version:

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JOURNAL DE PHYSIQW CoZZoque C8, suppZe'ment au n08, Tome 41, aoct 1980, page C8-586

CALCULATION OF THE PARTIAL STRUCTURE FACTORS I N BINARY L I Q U I D ALLOYS BY A THERMODYNAMICAL APPROACH

M. Favre-Bonte, J . C . Joud, P. H i c t e r and P . Desre

~ a b o r a t o i r e de Thermodynamique e t Physico-Chimie Me'taZZurgiques, associe' au C.N.R.S., ENSEEG, Domaine U n i v e r s i t a i r e , B.P. 44, 38401 S a i n t Martin dlHe'res, France.

INTRODUCTION : t i c a l diameters ( " s u b s t i t u t i o n a l hypothesis"), The s t r u c t u r e of t h e b i n a r y l i q u i d a l l o y s hence p=pl. From (1) and (2) equations, we o b t a i n can be h a r d l y determined experimentally. A complete t h e simple r e l a t i o n s :

i n f o r m a t i o n o f t h i s s t r u c t u r e can be reached o n l y ai j ( k ) = a M ( k ) + 4 n p r (gi (r)-gM(r))-, r2 d r (3)

0

f o r p a r t i c u l a r cases. Therefore, i t i s i n t e r e s t i n g We i n t r o d u c e : g. . ( r ) = a . .(xA,r) gM(r), where 1 J 1 J

t o determine t h e o r e t i c a l l y t h e p a r t i a l s t r u c t u r e aij(xAYr) i s a f u n c t i o n d e s c r i b i n g b o t h s i z e e f f e c t s f a c t o r s (PSF). I n general, t h e PSF obtained by t h e and chemical e f f e c t s and xA i s t h e atomic c o n c e n t r a Percus-Yerick hard sphere model are n o t s a t i s f a c t * t i o n of t h e species A. These d i f f e r e n t f u n c t i o n s r y because chemical e f f e c t s a r e n o t taken i n t o ac- a r e n o t independent and t h e f o l l o w i n g r e l a t i o n can count. We have a l r e a d y proposed a simple method be e a s i l y determined :

t o determine d i r e c t l y t h e PSF from thermodynamic (aAA(xA9r)-1) =

-

T ( ~ A M ( ~ A Y r ) - 1 ) = * ( % ( ~ A y ~ ) - 1 X 4 ) X~ A x ~ 2

p r o p e r t i e s o f t h e m i x t u r e . T h i s method c o n s i s t s i n where xM i s t h e atomic c o n c e n t r a t i o n o f t h e s o l v e n t c o r r e c t i n g t h e s t r u c t u r e f a c t o r of a s o l v e n t m a t r i x M. If t h e l i q u i d a l l o y i s disordered, aij(xA,r)=l as modified by chemical i n t e r a c t i o n s . We have used and a l l t h r e e PSF are i d e n t i c a l . For an a l l o y w i t h a s t a t i s t i c a l quasichemical treatment based on t h e s t r o n g chemical i n t e r a c t i o n s , t h e parameters aij

"surrounded atom" e n t i t y i n o r d e r t o i n t r o d u c e are d i f f e r e n t from one and can be considered as a chemical e f f e c t s . F i r s t l y , we r e c a l l t h i s formalism measure o f t h e l o c a l chemical o r d e r e f f e c t s . To a and r e p o r t the PSF evaluated f o r t h e Aq-Li and f i r s t approximation, t h e chemical e f f e c t s are res- Pb-Li systems. t r i c t e d t o f i r s t neighbours (a. . ( x A , r ) = l f o r r>a),a

1 J

i s the d i s t a n c e between f i r s t neighbours defined by 1

-

QUASICHEMICAL APPROACH.

-

4ap g M ( r ) r dr=z

1:

2 ; z i s t h e number o f nearest neigh-

The s t r u c t u r e f a c t o r o f t h e pure l i q u i d M bours of an atom, taken equal t o 10 i n t h i s case. and t h e FABER-ZIMAN PSF o f a AM l i q u i d a l l o y w i t h Moreover any v a r i a t i o n o f a .

.

(xA,r) w i t h d i s t a n c e

1 J

A d i l u t e species are given by : r i s neglected f a r O<r<a, consequently m _{s i n k r 2}

aM(k) = 1 + 4 7 ~ 1 (9M(r)-1),T r d r (1) ai j ( ~ A , r ) = a (xA). The parameter aAA(xA) can be

0 i j

6

s i n k r 2

a. .(k) = 1+4ap1 , ( g i j ( r ) - l ) T , r d r (i y j = A , ~ ) ( 2 )

evaluated from a s t a t i s t i c a l thermodynamic t r e a t - 1 J

where p and p ' a r e t h e average atomic d e n s i t i e s ment using t h e "surrounded atom" model. Such a treat- o f t h e l i q u i d M and t h e A-M a l l o y r e s p e c t i v e l y ; ment a l l o w s t o c a l c u l a t e t h e d e v i a t i o n AxA o f t h e g ( r ) i s t h e p a i r d i s t r i b u t i o n f u n c t i o n o f M and l o c a l composition from t h e mean value xA :

M ~ ( 1 - x A )

-

p

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

-

z

,

where

7

i s t h e average nun-

1 3

i n t h e A-M a l l o y . We assume t h a t A and M have iden- a A.

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aAA(xA) can be e a s i l y expressed as a f u n c t i o n of 2

AxA. I n f a c t , t h e i n t e g r a l 4npxAIa g A A ( r ) r d r equals

0

t h e number nA o f atom A surrounding an atom A.

-

Since gAA(r)'aAA(xA)gM(r) S nA=xAzaAA(xA) = z - ~ .

F i n a l l y we o b t a i n t h e r e l a t i o n :

The FABER-ZIMAN PSF f o l l o w from ( 3 ) ( 4 ) and ( 5 ) and a r e given, i n f i r s t order, by : s i n k r 2 aAA(k) = aM(k) t M dr (6) axA a _{s i n k r ,2} _dr aAM(k) = ay(k)

-

4 ~ -

I

gM(r) _kr X~ 0 ( 7 ) s i n k r 2 aMM(k) = aMfk) + 4 n ~ x A gFl(r) dr ('1

F o r t h e systems AgLi and Pb-Li, RUPPERSBERG e t a 1 ( 2 ) ( 3 ) have determined t h e PSF from X-Ray and neutron d i f f r a c t i o n ; u s i n g (6), (7) and ( 8 ) we have c a l c u l a t e d t h e PSF f o r Ag-Li a t xAg=0.285 and f o r - Pb-Li a t xpb=0.38. I n p a r t i c u l a r t h e f i r s t peak o f a l l t h r e e measured PSF was found t o be a t t h e same p o s i t i o n . This p a r t l y j u s t i f i e s t h e s u b s t i t u - t i o n a l hypothesis f o r these a l l o y s . The values of chemical l o c a l order parameter r e s u l t i n g from t h i s treatment a r e AxAg= -0.0373 f o r Ag0.285-Li0.715

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(using measurements by BECKER )and Axpb= -0.1155 f o r Pb0.38-LiD.62 (evaluated from thermodynamic values o f DEMIDOV e t a ~ ( ~ ) ) . The s t r u c t u r e f a c t o r o f t h e L i m a t r i x has been measured a t 593K by RUPPERSBERG and E G G E R ( ~ ) . I n f i g u r e 1 and 2 we can- pare c a l c u l a t e d and experimental values o f a . . (k)

.

1 J

The prepeaks o f t h e experimental PSF aAgVAg(k) and apbapb(k) demonstrate s u b s t a n t i a l contacts of Ag and L i , o r Pb and L i , r e s p e c t i v e l y due t o s t r o n g heteroatomic ordering. They a r e s t r o n g e r than those c a l c u l a t e d , hence t h i s chemical o r d e r e f f e c t i s o n l y p a r t i a l l y found i n t h e c a l c u l a t e d PSF. On t h e o t h e r hand t h e v a l u e s o f a. .(o) deduced t o

1 J

(6) (7) and (8) are i n r e l a t i v e l y good agreement

0 2 4 6 K ~ I

F i g . 1 : L i q u i d AgoSzB5-Li

-

0.715 a t 593K.

--

(a)

a Ag.Ags ( b ) aLi-~i ( c ) a A g - ~ i

.-

(a) a A g . ~ g +

1.06sNC, (b) aLi-Li-8.81SNC, (c) aAg.Li-3.81SNC.

Fig. 2 : L i q u i d PbD.38-Li0.62

---

aij(k) a t 593K,

-

a. . ( k ) a t 773K : ( a ' ) a p b - p b ~ ( b ' ) aLi-Li

,

1 J

(''1 apb-Li'

w i t h those evaluated from thermodynamic data as shown i n f i g u r e 1 and 2. T h i s agreement i s expec-

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C8-588 JOURNAL DE PHYSIQUE

should not be understood as the real deviation i n the s h e l l i . We obtain s i m i l a r r e l a t i o n s f o r aA&k) concentration from t h e s t a t i s t i c a l mean value j u s t

a t the nearest neighbour distance. I t represents, instead, the deviation i n concentration of the "mean 'field" type as averaged over longer distances. Obviously AxA, as defined t h i s way, cannot r e s u l t i n the correct experimental prepeak i n t e n s i t y , since t h e prepeak r e f l e c t s a superstructure due t o an alternation of layers enriched in atoms A and M respective1 y. Hence, t h i s superstructure e f f e c t should be explained by extending chemical order e f f e c t s . Therefore the quasichemi cal model wi 11 not be used f u r t h e r .

2

-

CHEMICAL ORDER DAMPING EFFECT.-

In order t o include chemical order e f f e c t s beyond the f i r s t neighbours we subdivide the space surrounding a central atom A by spherical s h e l l s and a d j u s t t h e i r thickness t o t h e minima of t h e p a i r d i s t r i b u t i o n function of t h e matrix M , outside

the t h i r d neighbours we choose, a s period, t h e thickness of t h e t h i r d layer. In t h i s way, we defi- ne, f o r each s h e l l i (i=1,2,

...

n ) , the deviation of the mean concentration xA of the species A by :

AX ( i ) = x (jl-xA. Since the radial decay function A A

of the chemical order parameter is not known we choose, ,?or the sake of mathematical simplicity, a l i n e a r law : AxA('*')= - a o i s the so cal- led chemical order damping c o e f f i c i e n t (CODC) which i s positive. The negative sign r e f l e c t s a pe- r i o d i c arrangement of the species A and M , on the

and aMM(k). For k=O, r e l a t i o n (12) becomes very simple :

where zi i s the number of atoms in the s h e l l i . Without the knowledge of o and bxA(ll in the f i r s t s h e l l , we cannot d i r e c t l y use r e l a t i o n (12). In order t o estimate t h e CODC and b ~ ~ ( ~ ) we use ( i n eqla- tions (12) and (13)) the experimental value of the prepeak amplitude aAA(k) and the thermodynamic l i - mit aAA(0). This treatment was applied t o the l i - quids Ag0.285-Li0. 715 and Pbo.38-Li0.62. On t a b l e I a r e reported t h e values of a and bxA(') f o r various

n

values from 4 t o 9. We l i m i t t h e analysis of 9 layers which approximately correspond t o a distance

0

of 25 A from the central atom. Further the micro- structure reflected by the o s c i l l a t i o n s of

2

r ( g .

.

( r ) - 1 ) p r a c t i c a l l y disappears(6). Moreover

1 J

since aAA(0) represents the thermodynamic l i m i t , an calculation

m u s t

include a s u f f i c i e n t l y high number of s h e l l s and consequently, r e s u l t s are given f o r

n

values greater than 4 (corresponding t o about 10

W)

only.

. .

average. The extension of t h e previous formulation a1 joys _Li0.715 _{and Pb0.38 Li0.62} n 4 5 6 7 8 9

of the PSF f o r

n

s h e l l s can be d i r e c t l y written : Note t h a t t h e CODC and bxA(l) a r e almost independent AxA( 1)

aAA(k)=aM(k)t4a of t h e number n of s h e l l s . This r e s u l t s in the rela-

.

P x A

t i v e l y small value of a'which rapidly damps the c h e n

x t

[J+'

s i n kr 2

i =l g r dr mica1 order and implies t h a t the contribution of the

Table I : Values of CODC and b x A ( l l f o r the l i q u i d

where ai and ai,r a r e the inner and

outer

r a d i i ,

of

i n t e g r a l s from t h e f i f t h shell Can be neglected. I t

Ag0.285 Li0,.715

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t h e absolute values o f

~ x ~ ( ' ) = - 0 . 1 4 4 , ~ b ~ . ~ ~ - ~ i 0.62 bxA(ll= -0.22) a r e much g r e a t e r than those deduced from t h e quasi-

p l i e s f o r aAM(k) and aMM(k) which i n c o n t r a s t t o aAA(k) have n o t been adjusted.

chemical model

.

BLETRY' s geometrical hard sphere model ( 7 ) , when appl l e d t o Ag0.285-Li0.115 leads t o a A x A ( l ) of =

-

0.17 which i s c l o s e t o t h e one we estimated above. This agreement supports our model, i n p a r t i c u l a r t h e estimated a which show t h a t about 80 % o f t h e chemical o r d e r i s l o c a l i z e d t o t h e f i r s t she1 1. The chemical o r d e r can be neglec- t e d a t i n t e r a t o m i c d i s t a n c e s g r e a t e r than 4. From

AX^(')

and CODC, we have c a l c u l a t e d t h e PSF. a. . ( k ) 1 J are p l o t t e d i n f i g u r e 3 and 4.

7 -

-

I

Fig. 4 : L i q u i d Pb0.38-Li0.62 ( a ' ) a p b - p b ~ ( b ' )

-

aLi-Lj

'

(''1 a p b - ~ i CONCLUSIONS : F o r a d i l u t e s u b s t i t u t i o n a l a l l o y w i t h s t r o n g heteroatomic i n t e r a c t i o n s , t h e knowled- 0 2 4 6 Ki-I

Fig. 3 : L i q u i d Ag0,285-Li0.715 (a) aAgaAg, (b)

-

a ~ i - ~ i ' ('1 aAg-Li

The t h r e e c a l c u l a t e d PSF a r e s 1 i g h t j y o u t o f phase w i t h the experimental values. T h i s displacement i s probably due t o t h e departure from t h e p e r f e c t s&s- t i t u t i o n a l behaviour o f these a1 l o y s . I n f a c t from t h e d i f f r a c t i o n measurements, RUPPERSBERG e t a1 have found t h a t t h e d i s t a n c e between u n l i k e n e a r e s t neighbours i s s m a l l e r t h a n t h e mean corresponding distances o f t h e pure components. The shape o f a l l t h e PSF i s f a i r l y w e l l represented. T h i s even ape

ge o f thermodynamic p r o p e r t i e s o f t h e m i x t u r e as w e l l as t h e s t r u c t u r e f a c t o r of t h e m a t r i x a l l o w s t o c a l c u l a t e t h e PSF u s i n g a thermodynamical approach. We have shown t h a t an extension of chemical o r d e r beyond f i r s t neighbours p e r m i t s t o expl sin e s s e n t i a l experimental featuresof t h e PSF. The proposed approach i s s e l f -consi s t e n t concerning expe- r i m e n t a l PSF

,

thermodynamic ropert ti es and a geome- t r i c a l model

.

REFERENCES :

(11 \ - I P. LATY. J.C. JOUD, J.C. MTHIEU, P. DESRE.

-.

~ h i l ' .. ..

Mag. B,-38, 1 (1978).

(2) H.

REITEK

H.

RUPPERSBERGj; kl. SPEICHER, 3 r d I n t . Conf. on L i a u i d iuletals, 1313 l B r i s t o l 1976). (4) W. BE;KER,'T~~S~ s U n i v e r s l t l t d e r Saarl ander

-

SaarbrLicken.

(5) A. I. DEMIDOV

,

A.G. MORACHEVSKII

,

L.N. GERASRENKO S o v i e t Electrochim., 9, 813'

f

1974).

(6) M.C. BELLISSENT-FUNEL: P. DESRE, Phi 1

.,

Mag., 36, 1063 (1977).