NON-ADDITIVE HARD SPHERE MIXTURES, NUMERICAL RESULTS FOR NEGATIVE NON-ADDITIVE PARAMETER

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NON-ADDITIVE HARD SPHERE MIXTURES,

NUMERICAL RESULTS FOR NEGATIVE

NON-ADDITIVE PARAMETER

D. Levesque, J. Nixon, M. Silbert, J. Weis

To cite this version:

JOURNAL DE PHYSIQUE ColZoque C8, suppZdment au n08, Tome 41, ao2t 1980, page

C8-317

N O N - A D D I T I V E HARD SPHERE M I X T U R E S , NUMERICAL RESULTS FOR N E G A T I V E N O N - A D D I T I V E PARAMETER,

D . Levesque, J.H. ~ i x o n * , M. ~ i l b e r t * and J.J. Weis

Laboratoire de Physique The'orique e t Hautes Energies. Orsay, ~ r a n c e ( + ) .

*

_{SchooZ of Mathematics and Physics, University of East AngZia, N o d c h , U.K.}

Abstract.- The Percus-Yevick approximation i s solved numerically f o r a binary mixture of hardspheres with negative non-additive diameter. Results f o r t h e p a r t i a l d i s t r i b u t i o n functions, the p a r t i a l s t r u c t u r e f a c t o r s and a few thermodynamic p r o p e r t i e s a r e presented. We discuss t h e possible usefulness of t h i s model a s a reference system f o r binary l i q u i d a l l o y s with s h o r t range order.

1. I*+u~t_izn_ I n t h e i r pioneer work on l i q u i d Cu-Sn a l l o y , Enderby e t a 1

[

11

pointed out t h a t , whereas t h e f i r s t peaks of t h e Cu-CU and Sn-Sn p a r t i a l s t r u c t u r e f a c t o r s corresponded c l o s e l y t o t h e pure component behaviour, t h e p o s i t i o n of t h e f i r s t peak of t h e Cu-Sn p a r t i a l s t r u c t u r e f a c t o r did not f a l l midway between t h e o t h e r two. The authors s t a t e d a t t h e time t h a t "it remains t o be seen whether t h i s departure from complete random mixing i s c o n s i s t e n t with t h e short-range order p o s t u l a t e d t o explain t h e thermodynamic p r o p e r t i e s of t h i s a l l o y system".

The simplest system which could e x h i b i t t h i s type of behaviour c o n s i s t s of a binary mixture of hard-spheres such t h a t t h e e f f e c t i v e diameter between spheres of unlike species i s given by

R 1 2 = & ( R ~ + R ~ ) + a ( 1 where R . i s t h e diameter o f species i and a t h e non-additive parameter.

I n a sense a b i n a r y mixture of non-additive hard spheres (IiAHS) i s constructed i n such a way a s t o model t h e tendency f o r segregation, when

a > 0 , and of compound forming o r chemical ordering when a < 0 . I n t h e l a t t e r case we have t h e following arrangement

i . e . t h e i n t e r a c t i o n between spheres of d i f f e r e n t species i s "encouraged" t o t h e e x t e n t t h a t they a r e allowed t o p e n e t r a t e each o t h e r . One

important consequence of t h i s overlap i s t o reduce t h e excluded volume of t h e system, v i z t o reduce i t s e f f e c t i v e packing f r a c t i o n . The way t h i s a f f e c t s t h e s t r u c t u r e and some of the thermo- dynamic p r o p e r t i e s of t h e system a t d i f f e r e n t concentration i s s t u d i e d f o r a p a r t i c u l a r case i n t h e p r e s e n t work. A more d e t a i l e d and systematic study w i l l be published elsewhere.

2. Res,ults Our c a l c u l a t i o n s were c a r r i e d out i n t h e Percus-Yevick

(PY)

approximation. _{It i s} worth noting t h a t f o r NAHS, the a n a l y t i c s o l u t i o g of t h e PY equation i n one-dimension agrees with t h e exact s o l u t i o n only up t o 0 ( a 2 ) [ 2

1.

The appropriate equations were solved numerically using a very f a s t i t e r a t i v e method developed by

Abernethy and G i l l a n

[

3

1.

To s p e c i f y t h e NAHS f o u r parameters a r e required i n a d d i t i o n t o t h e non-additive parametel

a. I n t h i s work we have chosen t o sneciPv t h e

(+) Laboratoire associ6 au Centre National de l a _{system by Bi, R2, c and}

_n;

_{c i s t h e concen-} Recherche S c i e n t i f i q u e .

% r a t i o n of species 2, such t h a t c l = 1

-

c , and

C8-318

JOURNAL DE PHYSIQUE

n

t h e t o t a l packing f r a c t i o n

n

= ( 1

-

c ) n i + cn2, w i t h q i t h e packing f r a c t i o n o f s p e c i e s i i n t h e pure s t a t e .

We assume t h a t t h e h a r d s p h e r e d i a m e t e r s remain unchanged w i t h c o n c e n t r a t i o n and such t h a t R1

/%

= 0.83. I n a l l our cdcu.l.ations R2 i s

a r b i t r a r i l y t a k e n t o b e t h e u n i t of l e n g t h . The v a l u e s o f n used a t d i f f e r e n t c o n c e n t r a t i o n s a r e l i s t e d i n Table 1. The v a l u e s o f b o t h R1/R2 and

q correspond t o t h o s e used b y Hoshino and Young

1:

4

1

i n t h e i r s t u d y o f t h e e n t r o p y o f mixing of compound f o r n i n e l i q u i d b i n a r y a l l o y s .

The parameter a i s chosen a s f o l l o w s . We assume t h a t t h e c o n c e n t r a t i o n f l u c t u a t i o n s Scc(0)

1:

5

]

has t h e v a l u e s shown i n Table 1. The shape o f S c c ( 0 ) i s a r b i t r a r i l y chosen t o have t h e same q u a l i t a t i v e f e a t u r e s found i n compound forming a l l o y s ( s e e e . g . [6] )

.

The v a l u e s of a chosen a r e t h o s e reproducing t h e given v a l u e s o f S ( 0 ) and a r e

C C a l s o l i s t e d i n Table 1. The v a l u e o f s c c ( 0 ) given a t c = 0 . 2 i s , t o a l l p r a c t i c a l purposes, t h e s m a l l e s t we can o b t a i n w i t h t h e s p e c i f i e d v a l u e s o f R. and 11. A s m a l l e r v a l u e m a q b e o b t a i n e d i f a l a r g e r

n

i s used. Table 1. Packing f r a c t i o n s , n o n - a d d i t i v i t y --A-

parameter, c o n c e n t r a t i o n , p a r t i c l e and c r o s s term f l u c t u a t i o n s .Values i n p a r e n t h e s i s g i v e t h e c o r r e s - ponding a d d i t i v e r e s u l t s .

With t h e system now completely s p e c i f i e d we e v a l u a t e t h e p a r t i a l r a d i a l d i s t r i b u t i o n f u n c t i o n s g. .(I-), and t h e p a r t i a l s t r u c t u r e f - > t o r s S. . ( k )

..-I& . . 1 J

d e f i n e d by

[

7

1

s i n k r r 2 d r ( 2 )

where n i s t h e number d e n s i t y and 6ij t h e Kronecker d e l t a . g.

.

( r ) and S.

.

( k )

,

a t c = 0.2, a r e shown, 1 J 1 J r e s p e c t i v e l y , i n F i g u r e s 1 and 2. The v a l u e s of S.

.

( 0 ) a r e l i s t e d i n Table 2 . 1- J

Figure 1. P a r t i a l r a d i a l d i s t r i b u t i o n functions a t c = 0.2

Figure 2. P a r t i a l s t r u c t u r e f a c t o r s a t c = 0.2

The values of sNN(0) and s N c ( 0 ) a r e l i s t e d i n Table 1.

The long-wavelength l i m i t o f t h e q u a n t i t i e s defined by e i t h e r equation 2 o r by e q w t i o n s ( h ) ,

(5)

and

(6)

r e l a t e t o s e v e r a l thermodynamic properties of t h e system

[

5

1.

We s h a l l be con- cerned only with the d i l a t a t i o n f a c t o r defined by

where V denotes t h e volume of t h e system, and t h e compressibility f a c t o r

I n equation ( 8 ) , kg i s t h e Boltzmann constant, T t h e temperature and KT t h e isothermal compres- s i b i l i t y . The concentration dependences o f 6

and 8 a r e shown i n Figure

4.

Discussion I n Figure 1 one of t h e more interes-

I n f a c t g 1 2 ( r ) s t a r t s increasing slowly t o a = 0.53 and then goes down t o maximimum a t 5 2

-

Figure

4 .

Concentration dependence of the d i l a t a t i o n and compressibility f a c t o r s

JOURNAL DE PHYSIQUE

C8-320

o s c i l l a t e very weakly around 1.0. The r a t h e r small values of g.

.

( r ) a t t h e contact points a r e , i n our

1J

view, r e l a t e d t o t h e smaller e f f e c t i v e excluded volume due t o non-additivity. The f i r s t coordina- t i o n number f o r g 1 2 ( r )

-

t h e l a r g e r sphere s i t t i n g a t t h e o r i g i n , t u r n s out t o be zl r

4

( t h e equi- v a l e n t a d d i t i v e value i s zl 9 ) . It appears t h a t , on average, t h e l a r g e r sphere has four smaller ones overlapping with it and very l i t t l e c o r r e l a t i o n otherwise.

The minimum of s 1 2 ( k ) - Figure 2

-

i s a t t h e same p o s i t i o n a s t h e maximum i n S ( k ) , a t _{C C} kR2 = 6.75, and it i s f a i r t o conclude t h a t t h e r e l a t i v e l y l a r g e minimum i n s 1 2 ( k ) follows from t h e small value of Scc(0). We a l s o note t h e small value of S12(0) a t t h i s p a r t i c u l a r concentration, which has been a s s o c i a t e d by some authors [l'] with compound forming tendencies i n binary a l l o y s . As a general r u l e t h e values of s 1 2 ( 0 ) and s,,(o) become smaller t h e more negative i s a a t

a given concentration. S l l ( k ) and s g 2 ( k ) behave s i m i l a r l y t o a hard sphere S ( k ) with a smaller e f f e c t i v e packing f r a c t i o n . The maximum of S l l ( k ) i s a t t h e same p o s i t i o n a s t h e maximum of S N N ( k ) , a t kR2 9 7.36, possibly indica- t i n g t h a t most o f t h e p a r t i c l e f l u c t u a t i o n i s due t o t h e smaller spheres. We do not observe t h e pre- peaks found i n t h e work of Laty e t a 1 187 and t h e model c a l c u l a t i o n s of Bldtry r9.1. These prepeaks a r e p r e s e n t i n c a l c u l a t i o n s c a r r i e d out using an a t t r a c t i v e p o t e n t i a l between species 1 and 2 which does not compete with t h e hard core i n t e r a c t i o n

[lo].

This p o i n t r a i s e s a few questions which we hope t o i n v e s t i g a t e i n t h e f u t u r e .

The concentration dependence of t h e compres- s i b i l i t y f a c t o r seems t o agree q u a l i t a t i v e l y with t h e experimental '8 expected of compound forming systems [ll, 123 with a maxinnm a t c = 0.2 where S ( 0 ) has t h e smaller value. The concentration

C C

dependence of t h e d i l a t a t i o n f a c t o r 6 shows a l a r g e change a t c = 0.2 - corresponding t o volume

contraction

-

which i s a l s o i n q u a l i t a t i v e

agreement with experimental expectations [ll, 131.

Schirmacher [14] has r e c e n t l y concluded t h a t t h e s i z e of t h e volume contraction i n a system i s con- nected with i t s degree of i o n i c i t y . While our model cannot help t o decide on whether o r why a system becomes i o n i c , t h e r e s u l t s shown i n ~ i g k e

4

ao not appear t o lend support t o h i s a s s e r t i o n .

I n conclusion we believe t h a t NAHS could eventually become a s u s e f u l a reference system f o r mixtures a s hard spheres have been f o r s i n g l e component systems.

Acknowledgements We a r e g r a t e f u l t o M r Abernethy, D r Gillan, Dr Hoshino, and Professor Young f o r making t h e i r work known t o u s p r i o r t o publication. One of us !JHN) i s grateful t o t h e SRC f o r a graduate studentship, another (MS) g r a t e f u l l y acknowledges Professor Young f o r s t i m u l a t i n g discussions and valuable suggestions and The Royal Society f o r timely f i n a n c i a l support through t h e i r M o p e a n Exchange Scheme.

( * ) ~ a b o r a t o i r e associ6 au Centre National de la

Recherche S c i e n t i f i q u e .

1:

11

Enderby JE, North DM and Egelstaff PA, Phil Mag (1966) 961.

[

21 Lebowitz J L and Zomick D, J Chem Phys

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-

1

31 Abernethy GM and G i l l a n M J , Molec Phys

2

(1980). 839.

[:

41

Hoshino

en and

Young WH, J Phys F (1950). In

t h e p r e s s .

1

51 Bhatia AB, Liquid Metals 1976, Conf Series No. 30, R Evans and D Greenwood ( ~ d s ) ( I n s t of Physics, B r i s t o l and e on don) 1977, p 21.

1

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[

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[

83

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Mag. B

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(19781, 1.

1:

91 Bldtry J , Z . Naturforsch

33a

(1978), 327.

[lo]

Harder JM and S i l b e r t M. Unpublished r e s u l t s .

[ll] McAlister SP, Crozier ED and Cochran JF, The Properties of Liquid Metals, S ~ a k e n c h i ( ~ d )

(Taylor and Francis, s on don) 1973, g 445. r121 Ruppersberg H and Speicher W. Z. Naturforsch

- -

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113)

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Liquid Metals 1976, Conf S e r i e s No. 30.

R Evans and D ~ r e e i w o o d (Eds) ( I n s t o f s R y s i c s , B r i s t o l and e on don) 1 77, w 1 3.