ATOMIC TRANSPORTDIFFUSION IN LIQUID METALS AND SIMPLE LIQUIDS

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ATOMIC TRANSPORTDIFFUSION IN LIQUID METALS AND SIMPLE LIQUIDS

M. Gerl, A. Bruson

To cite this version:

M. Gerl, A. Bruson. ATOMIC TRANSPORTDIFFUSION IN LIQUID METALS AND SIMPLE LIQ- UIDS. Journal de Physique Colloques, 1980, 41 (C8), pp.C8-335-C8-344. �10.1051/jphyscol:1980885�.

�jpa-00220539�

JOURNAL DE PHYSIQUE CoZZoque C8, suppz&ment au n08, Tome 41, aoct 1980, page

C8-335

ATOMIC TRANSPORT.

DIFFUSION IN LIQUID METALS AND SIMPLE LIQUIDS

M. Gerl and A . Bruson

Laboratoire de Physique des SoZides, Uniuersite' de Nancy I , CO. 140, 54037 Nancy CGdex, France.

1

-

Ih'TROZ)uCTION

_where -f r i ( t ) denotes t h e p o s i t i o n of one o f t h e N A s good reviews o f t h e experimental measure- tagged p a r t i c l e s . For times l o n g compared w i t h t h e ments") and o f model c a l ~ u l a t i o n s ( ~ ) o f d i f f u s i o n c o l l i s i o n t i m e , G s ( r , t ) obeys t h e d i f f u s i o n equa- c o e f f i c i e n t s i n l i q u i d m e t a l s have been given i n t i o n ( 3 )

.

p r e v i o u s LAM c o n f e r e n c e s , t h e p r e s e n t paper w i l l _D V 2 ~ ~ ( r , t ) =

-

aGs ( 4 ) be mainly devoted t o t h e t h e o r e t i c a l d e s c r i p t i o n

a t

of t h e i r atomic t r a n s p o r t p r o p e r t i e s . Liquid metals whose s o l u t i o n corresponding t o a p a r t i c l e a t t h e a r e a c t u a l l y two component f l u i d s : t h e y can be o r i g i n a t t = I) can b e w r i t t e n :

c o n s i d e r e d a s made o f heavy p o s i t i v e i o n s immersed i n a b a t h o f conduction e l e c t r o n s . A good approxi-

1 2

G s ( r , t ) = exp (-

-

( 5 )

( ~ I T D lt1)312 .

. 4

D

l t l

mation t o t h e dynamics o f t h e i o n s i s t o assume

I n t h e same l i m i t , it i s p o s s i b l e t o o b t a i n t h a t t h e e l e c t r o n s f o l l o w them a d i a b a t i c a l l y ; t h i s

t h e i n t e r m e d i a t e s c a t t e r i n g f u n c t i o n : l e a d s t o a pseudo-atom d e s c r i p t i o n , where t h e t o t a l

energy o f t h e m e t a l can be w r i t t e n : F ( k , t ) = exp (-k 2 D t )

( 6 )

U = U

+ l

( 1 and t h e s e l f p a r t o f t h e dynamic s t r u c t u r e f a c t o r : 0 2

iij

Q ( r i j )

1 ^+co 1

sS(k,w) =,

1

~ ~ ( k , t ) eiwt d t =; ^{~ k ~ ( 7 )}

where U i s a l a r g e energy which depends s t r o n g l y o f _-00 w2 + ( ~ k 2 ) ~

t h e volume allowed t o t h e system, and ~ ( r ) i s a 2

which t a k e s t h e form o f a L o r e n t z i a n of w i d t h 2 Dk

.

two-body e f f e c t i v e p o t e n t i a l . A t c o n s t a n t volume,

S (k,w) a s measured by i n c o h e r e n t n e u t r o n s c a t t e - one may assume t h a t t h e dynamics o f t h e i o n s a r e

r:ng(4) on a mixture of s u i t a b l e i s o t o p e s , o r by governed by t h i s r e l a t i v e l y s h o r t ranged p o t e n t i a l

u s i n g p o l a r i z e d n e u t r o n s , should t h e r e f o r e begave and t h e r e f o r e , t h e dynamical d e s c r i p t i o n of l i q u i d

a s shown by t h e Eq. ( 7 ) n e a r t h e o r i g i n o f t h e m e t a l s should not d i f f e r s t r o n g l y from t h a t of

(k,w) p l a n e . m e g a u s s i a n f o m o f G s ( r , t ) i s . a l s o n e u t r a l monatomic l i q u i d s . I n t h i s review, we w i l l

v a l i d a t very s h o r t t i m e s ( f r e e p a r t i c l e behaviour) t h e r e f o r e be concerned with t h e dynamics of t h i s

and t h e r e f o r e a f a i r l y good approximation t o g e n e r a l c l a s s o f l i q u i d s .

~ ~ ( r , t ) a t a l l t i m e s : The d i f f u s i o n c o n s t a n t D measures t h e r a t e o f 312

v a r i a t i o n i n time of t h e mean s q u a r e displacement G ~ ( T , ~ ) = (

7

a ( t ) ) exp ( - r 2 a ( t ) ) ( 8 )

o f a tagged p a r t i c l e : k2 2

o r ~ ~ ( k , t ) = exp ( -

7

<r ( t ) > ) ( 9 )

< r 2 ( t ) > =

6

D t ( l a r g e t ( 2 )

An a l t e r n a t i v e e x a c t form f o r D can b e given hi^ is c l e a r l y a concept v a l i d f o r s u f f i c i e n t - i n terms o f $ ( t ) t h e v e l o c i t y a ~ t 0 ~ 0 r r e l a t i o n func- l y l o n g t i m e s ; f o r i n s t a n c e i f t i s s m a l l e r t h a n t h e t i o n (VAF). S t a r t i n g from t h e d e f i n i t i o n ( 2 ) o f D ,

- 1

i n v e r s e

r

o f t h e c o l l i s i o n frequency, t h e tagged We Can w r i t e : p a r t i c l e moves f r e e l y and (r 2 ( t ) > ^Q, t2. The l i n e a r

= lim

$ % [

^ds2 < q o ) (;. s2-s

1,

( v a r i a t i o n o f < r 2 > w i t h t i s only observed when t h e t-

tagged p a r t i c l e has experienced many c o l l i s i o n s w i t h where we have used the fact that $ ( t )

= - < q t

1 ).

bath p a r t i c l e s . A t any time t h e s t a t i s t i c s of t h e 3 1

a t 2 ) > does not depend on t h e o r i g i n o f time.

tagged p a r t i c l e s a r e d e s c r i b e d by t h e s e l f p a r t G

The VAF $ ( t ) measures t o what e x t e n t t h e system of t h e d e n s i t y - d e n s i t y c o r r e l a t i o n f u n c t i o n :

N r e t a i n s t h e memory o f t h e v e l o c i t y and it can b e

$ ( r , t ) =

<iil ^{b ( f}

^{+ :i(o) - :i(t) )> ( 3 )} shown from ( 1 0 ) t h a t : Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980885

JOURNAL DE PHYSIQUE

D =

J;

$ t ) d t ( 1 1 )

I n o r d e r t o emphasize t h e r e l a t i o n between t h i s d e f i n i t i o n o f D and t h e e q u a t i o n s

( 6 )

and ( 7 ) , l e t u s n o t i c e t h a t :

where

P . - ( t ) = e i Z . F ( t )

K

and Z k ( t ) = G ( t ) e i Z . g ( t )

i r e t h e s p a t i a l F o u r i e r t r a n s f o r a s o f t h e d e n s i t y and c u r r e n t o f t h e tagged p a r t i c l e ; (31, and j k a r e r e l a t e d by t h e e q u a t i o n o f c o n t i n u i t y :

and t h e r e f o r e :

Taking t h e v e c t o r

2

a l o n g t h e x a x i s we o b t a i n :

0

o r , i n F o u r i e r t r a n s f o r m with r e s p e c t t o t :

T h e r e f o r e

a r e s u l t c o n s i s t e n t w i t h t h e e q u a t i o n ( 7 ) . The d e f i n i t i o n of

D

i n terms o f @ ( t ) proves more f r u i t f u l t h a n t h e d e f i n i t i o n ( 2 ) . I n t h e pre- s e n t paper we w i l l g i v e a review o f d i f f e r e n t me- thods used f o r c a l c u l a t i n g D . One approach t o t h e d i f f u s i o n c o n s t a n t i s t o w r i t e t h e k i n e t i c equa- t i o n s governing t h e d e n s i t y of t h e tagged p a r t i c l e . The Boltzmann e q u a t i o n , a p p r o p r i a t e t o d i l u t e gases, c o n s i d e r s t h a t t h e c o l l i s i o n s a r e independent, l o c a l i n time and s p a c e , and w e l l s e p a r a t e d . An improve- ment on t h i s e q u a t i o n i s t h e Enskog approximation, where t h e g e o m e t r i c a l c o r r e l a t i o n s between p a r t i c l e s a r e c o n s i d e r e d , b u t t h e c o l l i s i o n s a r e s t i l l assumed independent. These t h e o r i e s l e a d t o a simple expo- n e n t i a l decay o f t h e VAF whereas molecular dynamics c a l c u l a t i o n s show t h a t J, ( t ) has a r a t h e r complicated s t r u c t u r e . This s t r u c t u r e a r i s e s from d y n a n i c a l cor- r e l a t i o n s which a r e t a k e n i n t o account i n t h e theo-

r i e s we w i l l d e s c r i b e i n S e c t i o n 11. I n p a r t i c u l a r , t h e VAF becomes n e g a t i v e a f t e r a r e l a t i v e l y s h o r t time i n high d e n s i t y f l u i d s and e x h i b i t s a l o n g time t a i l due t o t h e coupling o f t h e tagged p a r t i c l e t o t h e hydrodynamic modes o f t h e f l u i d . The f i r s t pro- p e r t y o f $ ( t ) can b e u n t e r s t o o d by u s i n g a genera- l i z e d Langevin e q u a t i o n t o d e s c r i b e t h e s h o r t t i m e dynamics. We w i l l show i n S e c t i o n I11 how simple approximations t o t h e memory f u n c t i o n e n t e r i n g t h i s e q u a t i o n l e a d t o an o s c i l l a t o r y VAF. F i n a l l y t h e l o n g time t a i l of t h e VAF can be e a s i l y e x p l a i n e d u s i n g hydrodynamics o r g e n e r a l i z e d hydrodynamics, a s shown i n S e c t i o n I V .

I ?

-

KINETIC EQUATIONS

2. 1 .

-

The

BoRtztnann and

E v ~ ~ k o g - g ~ g t i ~ g

...

We c o n s i d e r h e r e t h e o n e - p a r t i c l e phase space d i s t r i b u t i o n f u n c t i o n fs(;,:,t) which g i v e s t h e d e n s i t y of t h e tagged p a r t i c l e s having t h e v e l o c i t y

+ +

v and l o c a t e d a t r a t time t . I n a d i l u t e g a s it i s p o s s i b l e t o c o n s i d e r o n l y b i n a r y c o l l i s i o n s , w e l l s e p a r a t e d i n time. The r a t e o f change i n time of

f s ( r , v , t ) + + i s given by t h e w e l l known e q ~ a t i o n ( ~ - ~ ~ ) :

-+

-+ -f

afs

3 f ( r , v , t ) + v.

7

t s

a

r = n c S ( ' ) f s ( ~ , ~ , t ) (1.0 where C S ( l ) i s t h e o p e r a t o r d e s c r i b i n g t h e d e t a i l e d mechanics o f t h e c o l l i s i o n s

,

and n i s t h e number d e n s i t y o f t h e f l u i d . By F o u r i e r t r a n s f o r m a t i o n of ( 1 7 ) we o b t a i n f o r each component f :

s q

a,

fsq = A f s q with

A =

-

i q v X

+

n

cs

( 1 ) ( 1 8 ) where we assume t h a t q p o i n t s along t h e -+ x a x i s . It i s i n p r i n c i p l e p o s s i b l e t o determine t h e eigen- f u n c t i o n s J, and t h e e i g e n v a l u e s o f t h e opera-

n

t o r A appearing i n t h e r. h. s . of t h e Eq.(18) :

A c t u a l l y , i n t h e l i m i t o f l o n g t i m e s , n

= I *

^{dv f} G , t ) obeys t h e d i f f u s i o n e q u a t i o n :

s q s q

and t h i s e q u a t i o n i s a l s o v e r i f i e d by f ( v , t ) . + s q

This t e l l s us t h a t t h e o n l y e i g e n v a l u e r e l e v a n t t o t h e s e l f - d i f f u s i o n problem i s AOq which v a n i s h e s a s q-m _and

I n t h e l i m i t o f s m a l l q t h e o p e r a t o r - iqvx which appears i n A can b e considered a s a small per- t u r b a t i o n and it i s e a s y t o c a l c u l a t e

ttoq

u s i n g s t a n d a r d p e r t u r b a t i o n t h e o r y :

where t h e s c a l a r product i s d e f i n e d by

and 3/2 2

eq ( v ) = ( --2!-- ) exp ( -

-

⁾ ( 2 3 )

21~krkT 2kT

i s t h e Maxwell d i s t r i b u t i o n f u n c t i o n . I n t h e s e equa- t i o n s we have assumed ( i ) t h a t t h e d i s t r i b u t i o n f u n c t i o n o f t h e b a t h p a r t i c l e s i s n o t p e r t u r b e d by t h e presence o f t h e tagged p a r t i c l e and ( i i ) t h a t t h e p r o b a b i l i t y of c o l l i s i o n between two tagged p a r t i c l e s i s very s m a l l , s o t h a t t h e o p e r a t o r Cs ( 1 ) i s l i n e a r .

The complicated i n t e g r a l a p p e a r i n g i n t h e r . h s . o f t h e e q u a t i o n ( 2 1 ) can be c a l c u l a t e d u s i n g a v a r i a t i o n a l procedure. I n t h e lowest approximation, one o b t a i n s t h e simple r e s u l t :

where t h e c o l l i s i o n i n t e g r a l C 2 ( ' ) ( 1 ) depends on t h e i n t e r a t o m i c p o t e n t i a l . I n t h e simple c a s e of h a r d s p h e r e s o f diameter o, t h i s e x p r e s s i o n reduces t o :

This r e s u l t provides a u s e f u l d e s c r i p t i o n o f d i l u t e g a s e s (n a 3 << 1) b u t f a i l s t o be v a l i d as t h e d e n s i t y i n c r e a s e s , because many s i m p l i f i c a t i o n s

(10) have been i n t r o d u c e d i n t h e Boltzmann e q u a t i o n

.

( i ) only independent b i n a r y c o l l i s i o n s have been c o n s i d e r e d ;

( i i ) t h e s e c o l l i s i o n s a r e c o n s i d e r e d l o c a l i n space and time. When a c o l l i s i o n o c c u r s at p o i n t r , +-

-+ -+

t h e d i s t r i b u t i o n f u n c t i o n f s ( r , v , t ) h a s been used i n t h e c o l l i s i o n i n t e g r a l whereas one should u s e

-+

.+

f,($

-

A;,

3,t -

At) where r

-

A r i s t h e p o i n t t h e molecule i s coming from and A t i s t h e t i m e necessary

f o r - t h e molecule t o r e a c h t h e p o i n t +- r , from t h e pre- ceding c o l l i s i o n .

( i i i ) t h e p r o b a b i l i t y o f c o l l i s i o n of two mo- l e c u l e s i s r e l a t e d t o t h e two-body d i s t r i b u t i o n

+ +

f u n c t i o n f(Z)(:,

3

;

r ,

v r ; t ) . Tn t h e Boltzmann

e q u a t i o n it i s assumed t h a t :

This approximation i s very crude : it n e g l e c t s t h e geometcical c o r r e l a t i o n s between p a r t i c l e s , which a r e b u i l t up by t h e i r i n t e r a c t i o n s .

A g r e a t improvement over t h e Boltzmann r e s u l t ( 2 5 ) has been o b t a i n e d semi-empirically by Enskog

I ) . ~e t a k e s i n t o account t h e s e g e o m e t r i c a l cor- r e l a t i o n s by o b s e r v i n g t h a t t h e c o l l i s i o n r a t e i s n o t governed by n t h e d e n s i t y of b a t h p a r t i c l e s , b u t by n g ( o ) where g ( a ) i s t h e 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 (RDF) a t c o n t a c t . Therefore t h e b i n a r y c o l l i s i o n d i f f u s i o n c o e f f i c i e n t of h a r d spheres i n t h e Enskog approximation i s simply :

If one u s e s r e a s o n a b l e hard s p h e r e (HS) d i a m e t e r s a , t h i s simple r e s u l t p r o v i d e s v a l u e s of s e l f d i f f u s i o n c o e f f i c i e n t s which a r e o f t h e r i g h t o r d e r

(12)

of magnitude

.

2 . 2 . -

Denndg expannion oj Zhe dijdunion - - _ _ _ - _ _ - - - _ - --- --- _--- -_---

codd&Gc@

A s t h e d e n s i t y i n c r e a s e s , it i s no more v a l i d t o c o n s i d e r only u n c o r r e l a t e d b i n a r y c o l l i s i o n s . I n high d e n s i t y f l u i d s s u c c e s s i v e c o l l i s i o n s a r e high- l y c o r r e l a t e d . Some sequences o f such c o r r e l a t e d

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

I n f i g u r e 1 ( a ) a p r o c e s s by which t h e tagged p a r t i c l e ( 1 ) undergoes two s u c c e s s i v e u n c o r r e l a t e d c o l l i s i o n s i s d e s c r i b e d ; ( 1 ) c o l l i d e s f i r s t w i t h

( 4 )

and s u b s e q u e n t l y with p a r t i c l e ( 2 ) and p a r t i - c l e s ( 2 ) and ( 4 ) do n o t i n t e r a c t ( d i r e c t l y o r i n - d i r e c t l y ) w i t h one a n o t h e r when t h e t a g g e d p a r t i c l e ( 1 ) t r a v e l s between t h e space time p o i n t s A and B.

F i g u r e 1 ( b ) shows a sequence o f two c o r r e l a t e d col- l i s i o n s : a t t h e space time p o i n t B t h e tagged par- t i c l e c o l l i d e s a g a i n w i t h t h e p a r t i c l e ( 2 ) wit11 which it had c o l l i d e d b e f o r e . These c o l l i s i o n s a r e of t h e g e n e r a l t y p e d e p i c t e d i n f i g u r e 1 ( c ) : a f t e r t h e f i r s t c o l l i s i o n between ( I

1-

and (2) a t A : t h e b a t h p a r t i c l e e x p e r i e n c e s a sequence o f c o l l i s i o n s with o t h e r b a t h p a r t i c l e s ; i n t h e same time t h e tagged p a r t i c l e undergoes a s e r i e s o f , u n c o r r e l a t e d c o l l i s i o n s with b a t h p a r t i c l e s and c o l l i d e s a t B w i t h a b a t h p a r t i c l e ( n ) which has ( d i r e c t l y o r in- d i r e c t l y ) i n t e r a c t e d w i t h t h e i n i t i d b a t h p a r t i c l e

c8-338

^JOURNAL DE PHYSIQUE

( 2 ' ) . Notice t h a t p a r t i c l e ( n ) may be t h e same a s sequence d e s c r i b e d by c ~ ( ~ ) i s d e p i c t e d i n f i g u r e p a r t i c l e ( 2 ) , a s i n f i g u r e 1 (b). I n t h i s " r i n g cO1- 1 ( b ) . C o n s i s t e n t l y with t h e Eq. ( 29) we may t r y l i s i o n " t h e second c o l l i s i o n r e t a i n s t h e memory of to expand t h e d i f f u s i o n D in successive t h e f i r s t one and it i s c l e a r t h a t such c o r r e l a t e d powers o f t h e d e n s i t y n :

sequences must b e t a k e n i n t o account i n o r d e r t o

D = (D")

+

n D(2)

+

n2 D(3)+

. . .

^.) ( 3 0 ) p r o p e r l y d e s c r i b e s e l f d i f f u s i o n o r i m p u r i t y d i f f u -

s i o n .

Figune 1 :

Two successive c o ~ i o i o v l s

06

*he fagged pamXcLe ( 1

I vkniz

b& tx~~tLCee5

may

be uncom&ed

la),

oh coh- h&ed ( b and

d) .

f i g . 1 ( c ) : hing c o L l h i o n

A f i r s t s t e p toward t h e i n t r o d u c t i o n o f c o r r e - l a t i o n s c o n s i s t s i n a g e n e r a l i z a t i o n of t h e Eq.(21) which can be f o r m a l l y w r i t t e n ( 9 ) :

The s i m p l e s t g e n e r a l i z a t i o n o f t h i s e x p r e s s i o n i s :

Using a p e r t u r b a t i o n expansion, t h e f i r s t c o r r e c t i o n D ( ~ ) can be f o r m a l l y w r i t t e n :

The important p o i n t t o n o t i c e i s t h a t t h i s c o r r e c t i o n d i v e r g e s a s n% i n a Ed-system,There- f o r e t h e expansion (30) i s not l e g i t i m a t e . It i s p o s s i b l e t o f i n d simple arguments (') t o show t h e ( u n p h y s i c a l ) o r i g i n of t h i s divergence. L e t us c o n s i d e r f o r i n s t a n c e t h e diagram d e p i c t e d i n f i g u r e 1 (b) ; i t can b e shown t h a t t h e f r a c t i o n o f b i n a r y c o l l i s i o n s which g i v e r i s e t o a r e c o l l i s i o n o f t h i s type a t any l a t e r time d i v e r g e s i n a 2d- system. But a c t u a l l y b o t h p a r t i c l e s a c t a s a s c r e e n which e l i m i n a t e s long time r e c o l l i s i o n s s o t h a t t h e d i f f u s i o n c o n s t a n t t a k e s t h e form (14)

.

The p r o p e r way t o t a c k l e t h i s divergence pro- blem i s t o c o n s i d e r r i n g s (') which exhaust t h e c l a s s o f t h e most d i v e r g i n g diagrams. A t y p i c a l r i n g i s d e p i c t e d i n f i g u r e 1 ( c ) . T h e o r i e s which i n c l u d e , i n a d d i t i o n t o t h e Enskog t e r m , t h e s e r i n g e v e n t s

arq

k n w a s r i n g t h e o r i e s

'

15-19' and have been used t o o b t a i n t h e l o n g time t a i l of t h e v e l o c i t y auto- c o r r e l a t i o n f u n c t i o n (20

$*'

⁾ d i s c o v e r e d by molecu- l a r dynamics c a l c u l a t i o n s ( 2 2 ' , and t o c a l c u l a t e s e l f - d i f f u s i o n c o e f f i c i e n t s . Repeated r i n g s a r e a l s o i n c l u d e d by Mehaffey and C ~ k i e r ' ~ ~ ) i n t h e i r k i n e t i c t h e o r y o f s i n g l e - p a r t i c l e motion i n a f l u i d .

2.3. - Ring

and

Repeated Ring T h e o h i a

--- --- --- ---

D 1 L - L

The essence o f t h e c a l c u l a t i o n o f Mehaffey

s and Cukier i s a s follows. They d e f i n e t h e Laplace

with Cs

=

n C S ( ' )

+

n2 Cs(2)+

. . . . .

( 29) Transfoim of t h e VAF _{a s}

where c ~ ( ~ ) d e s c r i b e s a p r o c e s s i n v o l v i n g two b a t h $J(z)

" - I

^dl v l x Cs(12) V2x p a r t i c l e s . Among t h e s e p r o c e s s e s , some a r e uncorre- ni

fi

l a t e d and have t h e r e f o r e a l r e a d y been c o n s i d e r e d where ni i s t h e number o f tagged p a r t i c l e s p e r u n i t by i t e r a t i o n o f c s ( )

.

An example of c o r r e l a t e d volume, 52 i s t h e volume o f t h e system and Cs(12) i s

t h e Laplace Transform (LT) of t h e tagged p a r t i c l e d e n s i t y c o r r e l a t i o n f u n c t i o n :

They show t h a t Cs(12) obeys t h e well-known e q u a t i o n

where L o ( ? ) i s t h e tagged p a r t i c l e f r e e streaming L i o u v i l l e o p e r a t o r , Qs(13) i s t h e LT o f t h e me- mory f u n c t i o n and ?s(12) i s t h e e q u a l time c o r r e - l a t i o n f u n c t i o n . The e q u a t i o n o f motion of Qs i n v o l v e s h i g h e r o r d e r c o r r e l a t i o n f u n c t i o n s which must b e approximated i n some way b u t Qs can b e f o r m a l l y w r i t t e n :

where

EM

^{i s} t h e mean f i e l d v a l u e of t h e four-point c o r r e l a t i o n f u n c t i o n G (17, 22) d e s c r i b i n g t h e event where t h e tagged p a r t i c l e l o c a t e d a t t h e p o i n t 1 i n t e r a c t s f i r s t w i t h a b a t h p a r t i c l e a t

i

and l a t e r w i t h a n o t h e r ( o r t h e same) b a t h p a r t i c l e a t p o i n t

2.

Geometrical c o r r e l a t i o n s a r e i n c l u d e d i n

EM

b u t all dynamical e j f e c t s a r e n e g l e c t e d s o t h a t t h e f i r s t term i n t h e Eq. ( 3 5 ) c o n t a i n s t h e Enskog approxima- t i o n t o Qs. Vs i n t h e Eq.(37) r e p r e s e n t s t h e i n t e r - a c t i o n between t h e t a g g e d p a r t i c l e and b a t h p a r t i - c l e s . Due t o s c l e e n i n g and excluded volume e f f e c t s , t h e p a r t i c l e s i n t e r a c t by t h e p o t e n t i a l Of mean f o r c e ( - k ~ I n g ( r ) ) r a t h e r t h a n by t h e b a r e i n t e r - atomic p o t e n t i a l ~ ( r ) . I f one u s e s t h e Eq. ( 3 5 ) , n e g l e c t s t h e c o r r e c t i n g term 6QS and makes simple approximations t o

EM ,

one r e c o v e r s t h e Enskog t h e o r y i n which t h e tagged p a r t i c l e undergoes o n l y u n c o r r e l a t e d c o l l i s i o n s with b a t h p a r t i c l e s .

The c o r r e c t i o n s t o t h e Enskog t h e o r y a r e con- t a i n e d i n 6QS. They can be s e p a r a t e d i n two c l a s - s e s :

a) i n t h e r i n g approximation one w r i t e s :

where T and i t s t r a n s p o s e T~ r e p r e s e n t Enskog bina- r y encounters between t h e tagged p a r t i c l e and b a t h p a r t i c l e s . The e q u a t i o n ( 3 6 ) r e p r e s e n t s an i n i t i a l c o l l i s i o n between t h e tagged p a r t i c l e

a

and a b a t h p a r t i c l e

B

( T T ), t h e n some complicated i n t e r m e d i a t e propagation ( G ' ) and a t e r m i n a t i n g c o l l i s i o n ( T )

between a and 6 o r a n o t h e r b a t h p a r t i c l e which h a s i n t e r a c t e d dynamically w i t h 6 s i n c e t h e f i r s t col- l i s i o n ; -cT and ^T a r e t h e r e f o r e c o r r e l a t e d c o l l i - s i o n s . Using t h i s approximation, Mehaffey and Cukier c a l c u l a t e t h e VAF + ( t ) and t h e d i f f u s i o n c o n s t a n t D. o f h a r d s p h e r e s of diameter ^Ui and mass misin d i l u t e s o l u t i o n i n a bath of s p h e r e s w i t h diameter 0 and mass ms, i n t h e l i m i t oi >> Us

3

¹ where 1 i s t h e mean f r e e p a t h o f b a t h p a r t i c l e s . I f One defines t h e Enskog v e l o c i t y r e l a x a t i o n frequency X E :

where T T ' i s t h e tagged p a r t i c l e - b a t h p a r t i c l e

1 s

c o l l i s i o n frequency, one g e t s i n t h e Enskog appro- ximation :

and

DiE =

-

kT ( 3 9 )

mi X~

C l e a r l y t h e t i m e dependence o f J I ~ ( ~ ) i s n o t c o r r e c t e i t h e r a t . s h o r t o r l o n g t i m e s .

I n t h e r i n g approximation one g e t s i n s t e a d of ( 3 8 ) and ( 3 9 ) :

and vE i s t h e Enskog kinematic s h e a r v i s c o s i t y . The e q u a t i o n ( 4 0 ) e x h i b i t s t h e c o r r e c t l o n g time behaviour o f $ ( t ) b u t , when t h e t a g g e d p a r t i - c l e i s l a r g e (0.; >>

o

), t h e c o e f f i c i e n t o f t ^-3/2 depends on t h e r a t i o ( o i / o ) 2 whereas f l u c t u a t i n g

'J i

hydrodynamics p r e d i c t s t h a t , f o r

-

>> 1

,

t h i s c o e f f i c i e n t i s independent of oi. us Moreover, t h e Eq. ( 4 1 ) p r e d i c t s t h a t , a s r l / A i n c r e a s e s ,

R E E

Di

/

D. i n c r e a s e s and becomes n e g a t i v e when r , >

E' a r e s u l t which i s c l e a r l y unphysical.

L e t us n o t i c e however t h a t f o r t a g g e d p a r t i - c l e s which a r e not t o o l a r g e , t h e r i n g t h e o r y pro- v i d e s a good d e s c r i p t i o n o f $(t) and D. For i n s t a n c e F u r t a d o e t a1 ( 2 1 ) w r i t e t h e memory f u n c t i o n ~ ( t ) o f t h e VAF a s f o l l o w s :

JOURNAL DE PHYSIQUE

~ ( t ) = XE 6 ( t ) + 6 K ( t ) where rlE =

v

En s m s i s t h e Enskog v a l u e o f t h e f l u i d s h e a r v i s c o s i t y . The c o e f f i c i e n t o f t-3/2 i n where XE ~ ( t ) i s t h e Enskog c o n t r i b u t i o n . The p a r t $ ( t ) a g r e e s w e l l with t h e r e s u l t s of f l u c t u a t i n g of 6 ~ ( t ) due t o r i n g c o l l i s i o n s i s c a l c u l a t e d in hydrodynamics. For very l a r g e p a r t i c l e s , t h e second a quasi-hydrodynallli~ approximation : t h e c o n t r i b u - term i n D? representing t h e c o n t r i b u t i o n o f succes- t i o n o f t h e f i v e hydrodynamic states is s i v e c o r r e l a t e d c o l l i s i o n s dominates eyer t h e f i r a t

e x a c t l y and t h e non-hydrodynamic modes a r e appro- and t h e d i f f u s i o n c o e f f i c i e n t t a k e s on t h e Stokes xirnately t a k e n i n t o account. The VAF and t h e r a t i o form.

D/D c a l c u l a t e d u s i n g t h i s procedure e x h i b i t t h e E

same v a r i a t i o n w i t h d e n s i t y a s t h e computer simula- A g e n e r a l i z a t i o n o f t h e s e c a l c u l a t i o n s t o sys- t i o n r e s u l t s ( f i g u r e 2 ) tems w i t h continuous i n t e r a c t i o n p o t e n t i d s can be

found i n a s e r i e s o f papers by Sjijlander e t al (24- 26)

(C(S)

I11

-

GENERALIZED LANGEVlN EQUATION AND SZMPLE MODELS

0.6

L,

^,

^{[Pi y}

K i n e t i c t h e o r i e s a r e n e c e s s a r y t o u n d e r s t a n d t h e

0.4 -

1.0 d e t a i l s o f t h e p h y s i c s i n v o l v e d i n t r a n s p o r t pro-

0 2 V l V 0

c e s s e s . The main d i f f i c u l t y with them i s t h a t t h e y

0 0.8

(4

become very i n t r i c a t e when one t r i e s t o r e a l i s t i c a l -

2 4 6 8 1 0 1 2 s l y d e s c r i b e t h e motion o f a tagged p a r t i c l e i n a f l u i d . One may t h e r e f o r e t r y t o r e l y on phenomenolo- Figune 2 : g i c a l t h e o r i e s . The f i r s t s t e p i n t h i s d i r e c t i o n

-

i s ( a ) VeLocity a&ocohrr&~tLon duncfhn $ ( A ) i n a t o use t h e w e l l known Langevin s t o c h a s t i c t h e o r y ( 2 8 )

bmd bphet~e @iiid do&

di+jdehent

v&en od

t h e

dm&.(on y i J a )

y

=

0.0741

; ( b ) y =

0.2468

which d e s c r i b e s t h e motion of a l a r g e and massive ( c ) y

- ^0.462ti

^{; b = -} 2

^X E ^t.

p a r t i c l e ( t h e i m p u r i t y ) i n a b a t h of s m a l l e r and ( 6 ) ~ a t L i a t i o n

06

be&-diddunion ~0

VIDE ul.iAh

V/Vo l i g h t e r p a r t i c l e s . I n t h i s c a s e t h e decay o f t h e

(dmm

he-6. ( 2 1 )

1.

v e l o c i t y of t h e i m p u r i t y o c c u r s i n a much l a r g e r in particular the enhancement of D,Dg at inter- t i m e t h a n t h e c o l l i s i o n t i m e . It i s t h e r e f o r e con- mediate d e n s i t y i s o b t a i n e d , a s w e l l a s t h e l o n g v e n i e n t t o w r i t e t h e e q u a t i o n o f motion o f t h e t i m e t a i l o f $ ( t ) . These f e a t u r e s a r e a l s o a p p a r e n t impurity as ^:

i n t h e c a l c u l a t i o n of Mazenko ( 1 5 ' 1 6 ) who c o n s i d e r s

i. + K vx = F ( t ) o n l v hvdrodvnamic c o n t r i b u t i o n s . and i n t h e work o f _" " X

R s s i b o i s (19'20) who determines t h e c o n t r i b u t i o n o f where

-

^{is a} sternatic retarding force c o r r e l a t e d sequences of two b i n a r y c o l l i s i o n s b e t - X

a c t i n g on t h e i m p u r i t y and m F ( t ) i s a random f o r c e ween t h r e e p a r t i c l e s .

&ch i s assumed t o have t h e f o l l o w i n g p r o p e r t i e s : b ) As t h e s i z e of t h e tagged p a r t i c l e i n c r e a s e s ,

r e p e a t e d c o l l i s i o n s become more important because t h e b a t h p a r t i c l e s r e p e a t e d l y r e t u r n t o t h e tagged p a r t i - c l e . The r e p e a t e d r i n g approximation i s n e c e s s a r y i n o r d e r t o remove t h e unphysical divergence of D found i n t h e r i n g t h e o r y . Using t h i s approximation, Mehaffey and Cukier show t h a t t h e l o n g time beha- v i o u r o f + ( t ) i s :

and t h a t t h e diff'usion c o e f f i c i e n t can be w r i t t d n

( a ) it v a n i s h e s i n t h e average ; ( b ) it i s uncorre- l a t e d w i t h t h e v e l o c i t y ( < F ( t ) v X ( o ) >

=

0 ) ; ( c ) i t s c o r r e l a t i o n time vanishes :< F ( t ) F ( o ) > =

2 B 6 ( t ) . Let us n o t i c e t h a t F and K a r e n o t inde- pendent : i f we i n t e g r a t e t h e e q u a t i o n ( 5 2 ) , use t h e p r o p e r t i e s of

F

and e x p r e s s t h e f a c t t h a t 1/2 m<vx> 2 t e n d s t o kT/2 because t h e i m p u r i t y become t h e r m a l i - zed i n t h e l o n g t i m e l i m i t , we o b t a i n a r e l a t i o n between t h e s t r e n g t h o f t h e f l u c t u a t i n g f o r c e and t h e magnitude K o f t h e f r i c t i o n c o e f f i c i e n t :

B = kT K (46)

This e q u a t i o n i s a c t u a l l y a simple v e r s i o n o f t h e (44 f l u c t u a t i o n d i s s i p a t i o n theorem (29y30).

Because F ( t ) i s not c o r r e l a t e d w i t h vx, it i s s t r a i g h t f o r w a r d t o show t h a t $ ( t ) = < v x ( t ) v x ( o ) >

obeys t h e e q u a t i o n :

and t h e r e f o r e decays e x p o n e n t i a l l y . Using t h e Eq.

( 1 1 ) t h e d i f f u s i o n c o e f f i c i e n t becomes :

a r e s u l t known a s t h e E i n s t e i n formula.

I n t h i s simple approximation t h e v a l u e of D depends d i r e c t l y on t h e v a l u e o f t h e f r i c t i o n coef- f i c i e n t K ( ~ ' ) which i s r e l a t e d t o t h e i n t e g r a l Of

t h e f o r c e - f o r c e c o r r e l a t i o n f u n c t i o n . K h a s been c a l - c u l a t e d u s i n g v a r i o u s approximations with r e a l i s t i c p o t e n t i a l s m d experimental s t r u c t u r e f a c t o r s .

The p r e c e d i n g simple a n a l y s i s u n f o r t u n a t e l y s u f f e r s from t h r e e major d e f i c i e n c i e s :

( a ) $ ( t ) e x h i b i t s a cusp at t = 0 , when it should be rounded a s t h e VAF i s an even function o f time ;

( b ) J, ( t ) decays e x p o n e n t i a l l y whereas molecu- l a r dynamics c a l c u l a t i o n s show t h a t ( i ) $ ( t ) may become n e g a t i v e i n high d e n s i t y f l u i d s and ( i i )

$ ( t ) has a Long t i m e t a i l d e c r e a s i n g slowly i n ti- me a s t-3'2 i n 3d systems.

The g r e a t advantage o f t h e Langevin e q u a t i o n i s t h a t it can e a s i l y b e g e n e r a l i z e d u s i n g t h e Mori formalism (31'32). The v e l o c i t y \(t) a c t u a l -

.-

l y obeys a very complicated L i o u v i l l e e q u a t i o n which d e s c r i b e s a l l t h e dynamics o f t h e d i f f u s i o n p r o c e s s . Mori h a s shown however t h a t t h i s L i o u v i l l e e q u a t i o n

i s e q u i v a l e n t t o t h e g e n e r a l i z e d Langevin e q u a t i o n : t

G

₊

j

d r K ( t

-

r ) v - r ) = a ( t )

X 0

(49 ) where ~ ( t ) i s t h e memory f u n c t i o n and a ( t ) is a random f o r c e ( i t is a c t u a l l y an a c c e l e r a t i o n ) which i s o r t h o g o n a l t o ( i

.

e. u n c o r r e l a t e d w i t h ) t h e velo- c i t y . The VAF t h e r e f o r e obeys t h e simple e q u a t i o n :

which ,makes s u r e t h a t $(o) = 0 i f t h e memory func- t i o n i s given a simple f u n c t i o n a l form: Moreover, it can b e shown t h a t :

( a ) a ( t ) a l s o obeys a g e n e r a l i z e d Langevin e q u a t i o n :

t

+

1

d r M ( t

-

r ) a ( r ) = ~ ( t ) (51 )

0

where M i s t h e memory f u n c t i o n o f a and R i s a new f l u c t u a t i n g f o r c e , o r t h o g o n a l t o a and v ;

( b ) K ( t ) i s r e l a t e d t o t h e f o r c e - f o r c e c o r r e l a - t i o n f u n c t i o n :

s o t h a t :

i s t h e s q u a r e o f t h e E i n s t e i n frequency of t h e i m p u r i t y , i . e . t h e frequency it would have i f t h e b a t h p a r t i c l e s were k e p t f i x e d a t t h e i r i n i t i a l p o s i t i o n . From t h e Eq. (51) and (52), it i s easy t o show t h a t t h e e q u a t i o n of motion of K ( t ) i s :

( 5 4 )

The p r o c e s s l e a d i n g t o t h e Eq. (50) and (54) can be i t e r a t e d w i t h t h e hope t h a t , a s one goes f u r t h e r , t h e memory f u n c t i o n s decay on a s h o r t e r a d s h o r t e r t i m e s c a l e . The sequence of e q u a t i o n s can b e

'I,

s o l v e d by L a p l a c e t r a n s f o r m a t i o n . I f $ ( s ) i s t h e t r a n s f o r m o f $ ( t ) we g e t :

It i s c l e a r t h a t t h e f a r t h e r one goes i n t h e con- t i n u e d f r a c t i o n expansion (551, t h e b e t t e r i s t h e d e s c r i p t i o n o f t h e s h o r t time dynamics of t h e d i f -

f u s i o n p r o c e s s . Even t h e s i n g l e r e l a x a t i o n time ap- proximation, where one assumes a n e x p o n e n t i a l decay of K ( t ) ( o r e q u i v a l e n t l y , a white spectrum f o r

~ ( t ) :

may p r o v i d e an o s c i l l a t o r y $ ( t ) . According t o t h e Eq. (52), we g e t :

which, a f t e r Laplace i n v e r s i o n g i v e s :

JOURNAL DE PHYSIQUE

~ 8 - 3 4 2

where s+ and s- a r e t h e r o o t s o f t h e denominator of t h e Eq. ⁽⁵⁸ ) . These r o o t s have an imaginary p a r t i f

~ K ( o ) > cx 2 o r :

I n t h i s c a s e , t h e VAF i s g i v e n by t h e e x p r e s s i o n :

$ ( t ) = $ ( o ) e - i / e c c t [ c o s ~ t + g s i n ~ t ] ( 6 1 ) R

where R2 = 4

no2 -

cc ²

A s

no

i s known from t h e i n t e r a t o m i c p o t e n t i a l and t h e e q u i l i b r i u m s t r u c t u r e o f t h e f l u i d , t h e p r e s c r i p t i o n (60) i s very u s e f u l t o determine i f t h e VAF has an o s c i l l a t o r y behaviour o r n o t . The s i n g l e r e l a x a t i o n t i m e approximation has been used by Berne, Boon and Rice(33) t p i n t e r p r e t t h e mole- c u l a r dynamics d a t a o b t a i n e d by Rahman i n argon. I f one wants t o improve t h e d e s c r i p t i o n o f t h e s h o r t time dynamics, one h a s t o use more s o p h i s t i c a t e d forms o f ~ ( t ) ( 3 4 ) , a s f o r i n s t a n c e t h e gaussian approximation o f Singwi and Tosi ( 3 5 )

o r t h e approximation o f Martin and Yip (36' :

which i n v o l v e s t h e c o r r e l a t i o n f u n c t i o n o f t h e time d e r i v a t i v e & o f t h e a c c e l e r a t i o n .

The main d e f i c i e n c i e s of t h e p r e c e d i n g a n a l y s i s a r e t h e f o l l o w i n g :

( a ) it does n o t p r e d i c t a v a l u e f o r t h e d i f f u - s i o n c o n s t a n t b u t r a t h e r makes use o f it a s a para- meter i n t h e c a l c ~ l a t i 0 n S ;

( b )

ii

i s e s s e n t i a l l y v a l i d a t s h o r t o r interme.

d i a t e times.

A c t u a l l y t h e coherent s t a t e of t h e tagged par- t i c l e changes by e x c i t a t i o n o r a b s o r p t i o n o f exci- t a t i o n s i n t h e f l u i d , which decay slowly i n time.

This g i v e s r i s e t o t h e l o n g time t a i l o f t h e VAF whose F o u r i e r Transform behaves a s a

-

b w

'I2

when w+O. T h i s b e h a v i o u r i s n o t expected i n t h e con- t i n u o u s f r a c t i o n expansion ( ~ q . 5 3 . I n o r d e r t o o b t a i n t h e l o n g time behaviour, o f $ ( t ) and t o b r i d - ge t h e gav'between l o n g and s h o r t t i m e s , Bosse . e t a 1 (37) approximately d e s c r i b e t h e c o u p l i n g o f t h e tagged p a r t i c l e t o l o n g i t u d i n a l and t r a n s v e r s e cur- r e n t e x c i t a t i o n s . They c a l c u l a t e M(t) t o l e a d i n g

o r d e r by assuming simple decay p r o c e s s e s by which t h e tagged p a r t i c l e e x c i t e s s i n g l e c u r r e n t mo'des i n t h e l i q u i d through an average dynamical m a t r i x . Using t h i s procedure t h e y can s e p a r a t e l y c a l c u l a t e t h e c o n t r i b u t i o n s o f l o n g i t u d i n a l and t r a n s v e r s e ex- c i t a t i o n s . T h e i r r e s u l t s compare favourably with t h e d a t a o b t a i n e d by molecular dynamics c a l c u l a t i o n s i n argon and rubidium. The main f e a t u r e s of t h e VAF spectrum a r e w e l l reproduced ( f i g u r e ( 3 ) ) , and i n p a r t i c u l a r t h e peak a t w % Qo i n argon o r w % 1,5 Ro i n rubidium.

Figme

3 :

V d o c i t y a L L t o c o h r r W o n hpecttuun $(wl i n cotnpahinon l u i t h

Rahmaah

^'i MP daAa (dashed cwcve] and uLith &e ken& obRained cntitkin $he hingLe h&xa;tion h e apphoxicimcttion (SRTJ l,jnarn

bed

(37) j.

Let us n o t i c e t h a t t h e e q u a t i o n ( 6 1 ) o b t a i n e d i n t h e s i n g l e r e l a x a t i o n t i m e approximation i s t y p i - c a l o f a damped harmonic o s c i l l a t o r : $ ( t ) e x h i b i t s o s c i l l a t i o n s a t s h o r t t i m e s ('r

g

10 -12 s ) and de- cays a t l a r g e t when d i f f u s i v e motions a c t u a l l y occur. A s a consequence t h e average p o s i t i o n -+ R o f t h e t a g g e d

article

changes much more slowly t h a n i t s a c t u a l p o s i t i o n + Ro. This remark l e d S e a r s ( 3 2 ) t o t h e i t i n e r a n t o s c i l l a t o r model where t h e t a g g e d p a r t i c l e v i b r a t e s i n t h e cage o f i t s n e a r e s t neigh- bours. The cage i t s e l f i s s u b j e c t e d t o a random f o r -

t

ce r f ( t ) and t o a s y s t e m a t i c r e t a r d i n g f o r c e - v R ( t ) . Using t h e e x p e r i m e n t a l v a l u e o f D and molecular dynamics d a t a , t h e spectrum of $ ( t ) he o b t a i n s f i t s reasonably w e l l t o t h e computer d a t a . Let us n o t i c e however t h a t t h i s a n a l y s i s i s n o t v a l i d a t s h o r t ti- me because t h e f o r c e s a r e n o t s t o c h a s t i c a t t h i s s c a l e o f t i m e . A s i m i l a r , q u a s i - c r y s t a l l i n e model

(391 h a s been used by Rahman e t al.

.

c o e f f i c i e n t s c o n t a i n an imaginary p a r t :

IV - HYDRODYNAMIC T H E O R Y

I n o r d e r t o t h e o r e t i c a l l y i n v e s t i g a t e t h e o r i - g i n of t h e l o n g time t a i l o f t h e VAF, it i s n a t u r a l t o r e l y on hydrodynamics. This h a s - b e e n done by Alder and Wainwright ( 2 2 ) who showed t h a t a t i n t e r - mediate d e n s i t y , t h e s o l u t i o n o f t h e Navier-Stokes e q u a t i o n l e a d s t o a hydrodynamic s t r u c t u r e of t h e v e l o c i t y f i e l d observed i n computer experiments on h a r d s p h e r e systems.

A s i m i l a r a n a l y s i s has been made by Zwanzig and Bixon ( 4 0 ) . According t o t h e Stokes-Einstein formula, t h e d i f f u s i o n c o e f f i c i e n t of a sphere of diameter i n a f l u i d with v i s c o s i t y 0 i s given by :

where

a

= 3 i f t h e f l u i d s t i c k s p e r f e c t l y t o t h e s u r f a c e o f t h e sphere and

a

= 2 i f t h e f l u i d s l i p s p e r f e c t l y o v e r t h i s s u r f a c e . Using experimental va- l u e s of D i n l i q u i d argon o r i n m e t a l s , t h e Eq.(64) l e a d s t o diameters

u

which a r e not unreasonable ( e s p e c i a l l y w i t h a = 2 ) . This remark g i v e s i t s f u l l value t o t h e a n a l y s i s of S t o k e s - E i n s t e i n , which can be improved i n two ways :

a ) I n t h e s t a n d a r d S t o k e s - E i n s t e i n t h e o r y t h e Navier-Stokes e q u a t i o n s a r e l i n e a r i z e d and, s t e a d y motion of t h e sphere i s assumed s o t h a t a l l d e r i - v a t i v e s a r e s e t equal t o z e r o . This amounts t o assu- ming t h a t t h e f l u i d i s incompressible, even i f i t i s a gas. It i s c l e a r t h a t , i n o r d e r t o d e s c r i b e t h e e r r a t i c motion of t h e sphere, one h a s t o extend t h i s t h e o r y t o t h e c a s e where t h e v e l o c i t y of t h e d i f f u s i n g s p h e r e v a r i e s w i t h time.

Using t h e same approximations a s b e f o r e (incom- p r e s s i b i l i t y of t h e f l u i d ) Stokes and Boussinesq g e t v a l u e s f o r t h e

VAF

which

do

n o t agree a t a l l w e l l with computer experiments.

b ) The reason f o r t h i s discrepancy can b e t r a - ced t o t h e inadequacy of t h e i n c o m p r e s s i b i l i t y ap- proximation. The time required f o r a sound wave t o propagate over an i n t e r a t o m i c d i s t a n c e i s about

s when t h e time s c a l e f o r d i f f u s i o n ( T % m/c) i s o f t h e same o r d e r o f magnitude. It i s t h e r e f o r e not l e g i t i m a t e t o n e g l e c t sound propagation. Moreo- v e r , it i s w e l l known t h a t f o r t h e high f r e q u e n c i e s

( 1 0 ' ~ s - l ) o f i n t e r e s t i n t h i s problem, a l i q u i d behaves v i s c c e l a s t i c a l l y . It i s t h e r e f o r e n a t u r a l t o assume t h a t t h e frequency dependent v i s c o s i t y

where K i s t h e v i s c o e l a s t i c r e l a x a t i o n time ( T %lo-13s i n l i q u i d a r g o n ) .

The l i n e a r i z e d f u l l Navier-Stokes e q u a t i o n s can then be s o l v e d , assuming t h a t t h e p r e s s u r e g r a d i e n t i s p r o p o r t i o n a l t o t h e g r a d i e n t o f t h e d e n s i t y .

Using r e a l i s t i c v a l u e s f o r t h e parameters ente- r i n g t h e t h e o r y , Zwanzig and Bixon o b t a i n a VAF

$ ( t ) and a frequency spectrum o f $ which compare favourably with t h e d a t a o f ~ a h m a i .

Let us a l s o mention t h e c a l c u l a t i o n o f Dorfnan ( 1 8 )

and Cohen

,

E r n s t e t a1.(41) who o b t a i n a de- t a i l e d e x p r e s s i o n f o r t h e l o n g time behaviour o f

$ ( t ) :

w h e r e v - n/nm i s t h e kinematic v i s c o s i t y . T h i s behaviour o f $ ( t ) has been checked on HS systems by Alder e t a l . ( 2 2 ) and by ~ e v e s q u e and Ashurst on Lennard-Jones f l u i d s (44)

VI - CONCLUSION

It t h i s review we have t r i e d t o show t h e d i f - f i c u l t y of t h e o r e t i c a l l y p r e d i c t i n g d i f f u s i o n coef- f i c i e n t s i n l i q u i d s . This d i f f i c u l t y a r i ~ e s essen- t i a l l y from t h e f a c t t h a t t h e d i f f u s i v e behaviour o f a tagged p a r t i c l e t a k e s p l a c e a f t e r t h e p a r t i c l e h a s experienced a g r e a t number of c o r r e l a t e d col- l i s i o n s . If t h e s e c o l l i s i o n s a r e c o n s i d e r e d indepen- dent from one a n o t h e r , t h e simple k i n e t i c models p r o v i d e d by t h e Boltzmann

-

Enskog e q u a t i o n s e a s i l y p r o v i d e v a l u e s o f d i f f u s i o n c o e f f i c i e n t s which a r e o f t h e r i g h t o r d e r of magnitude, b u t t h e d e t a i l e d dynamics a r e completely ignored. A s u s u a l i n s t r o n - g l y i n t e r a c t i n g many-body systems, p e r t u r b a t i o n ex- p a n s i o n s a r e v e r y d i f f i c a t t o perform because o f t h e c o r r e l a t i o n s between c o l l i s i o n s . A tagged par- t i c l e a t time t induces a backflow i n t h e f l u i d which r e a c t s b a c k on it a t a l a t e r t i m e , l e a d i n g t o memory e f f e c t s . C a l c u l a t i n g t h e e f f e c t o f t h e backflow i n d e t a i l s i s a very d i f f i c u l t t a s k b u t molecular dynamics c a l c u l a t i o n s have shed some l i g h t on some simple e f f e c t s .

C8-344 JOURNAL DE PHYSIQUE

There e x i s t t h e o r i e s which d e s c r i b e e i t h e r t h e 24 s h o r t time dynamics o f t h e tagged p a r t i c l e o r i t s 2 5 l o n g time behaviour. The d i f f i c u l t problem i s t o

b r i d g e t h e gap i n between e s p e c i a l l y with r e a l i s t i c 26 p o t e n t i a l s .

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