DYNAMICS OF SUSPENSIONS OF CHARGED PARTICLES

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DYNAMICS OF SUSPENSIONS OF CHARGED PARTICLES

R. Klein, W. Hess

To cite this version:

R. Klein, W. Hess. DYNAMICS OF SUSPENSIONS OF CHARGED PARTICLES. Journal de

Physique Colloques, 1985, 46 (C3), pp.C3-211-C3-222. �10.1051/jphyscol:1985317�. �jpa-00224634�

JOURNAL DE PHYSIQUE

Colloque C3, supplément au n03, Tome 46, mars 1985 page C3-211

DYNAMICS OF SUSPENSIONS OF CHARGED PARTICLES R.

and W. Hess

FakuZtüt fur Physik, Universitat Konstanz, 0-7750 Konstanz, F.R.G.

Résumé - On donne une d e s c r i p t i o n d e s p r o p r i é t é s dynamiques d e s s u s p e n s i o n s d e p a r t i c u l e s c o l l o ï d a l e s s p h é r i q u e s c h a r g é e s . On p a r t d ' u n e é q u a t i o n d e F o k k e r - P l a n c k e t on u t i l i s e l e f o r m a l i s m e de Mori-Zwanzig a f i n d ' o b t e n i r l e f a c t e u r d e s t r u c t u r e dynamique e t l a F o n c t i o n d e c o r r é l a t i o n d e c o u r a n t t r a n s v e r s e . On m o n t r e que c e s g r a n d e u r s d é p e n d e n t d e c o e f f i c i e n t s d e v i s c o s i t é du f l u i d e d e s p a r t i c u l e s c o l l o i d a l e s e n i n t e r a c t i o n , v i s c o s i t é q u i dépend de l a f r é q u e n c e e t du nombre d ' o n d e . La v i s c o s i t é e s t c a l c i i l é e d a n s l ' a p p r o x i m a t i o n du c o u p l a g e d e modes c e q u i ramène l e s p r o - p r i é t é s dynamiques du s y s t è m e au f a c t e u r d e s t r u c t u r e s t a t i q u e . On d i s c u t e p l u s i e u r s e x p é r i e n c e s q u i d é p e n d e n t d e l a v i s c o s i t é . En c e q u i c o n c e r n e l e s p r o p r i é t é s à u n e p a r t i c u l e , on p r é s e n t e d e s r é s u l t a t s s u r l e d é p l a c e m e n t q u a d r a t i q u e moyen e t l e c o e f - f i c i e n t d e d i f f u s i o n p r o p r e .

A b s t r a c t - The t h e o r y f o r t h e d y n a m i c a l p r o p e r t i e s o f c h a r g e d s p h e r i c a l c o l l o i d a l s u s p e n s i o n s i s d e s c r i b e d . S t a r t i n g from a Fokker-Planck e q u a t i o n , t h e Mori-Zwanzig f o r m a l i s m i s employed t o d e r i v e r e s u l t s f o r t h e dynamic s t r u c t u r e f a c t o r and t h e t r a n s - v e r s e c u r r e n t c o r r e l a t i o n f u n c t i o n . These q u a n t i t i e s a r e shown

t o depend on t h e f r e q u e n c y and w a v e v e c t o r d e p e n d e n t v i s c o s i t y f u n c t i o n s of t h e f l u i d o f i n t e r a c t i n g c o l l o i d a l p a r t i c l e s . The v i s c o s i t y f u n c t i o n s a r e c a l c u l a t e d i n a mode-coupling a p p r o x i - m a t i o n , which r e d u c e s t h e d y n a m i c a l p r o p e r t i e s of t h e s y s t e m t o

t h e s t a t i c s t r u c t u r e f a c t o r . S e v e r a l e x p e r i m e n t s , which depend on t h e v i s c o s i t y f u n c t i o n s , a r e d i s c u s s e d . With r e g a r d t o s i n g l e - p a r t i c l e p r o p e r t i e s , r e s u l t s f o r t h e mean-square d i s p l a c e m e n t and t h 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 a r e p r e s e n t e d .

1. INTRODUCTION

S u s p e n s i o n o f h i g h l y c h a r g e d s p h e r i c a l p a r t i c l e s h a v e b e e n i n v e s t i g a t e d i n r e c e n t y e a r s by s e v e r a l e x p e r i m e n t a l methods. The most w i d e l y u s e d method h a s p e r h a p s b e e n s t a t i c and q u a s i e l a s t i c l i g h t s c a t t e r i n g C l ] . But t h e r e a r e a l s o s e v e r a l n e u t r o n s c a t t e r i n g e x p e r i m e n t s on m i c e l l a r s y s t e m s 1 2 1 and on p o l y s t y r e n e s p h e r e s [ 3 ] . F i n a l l y , t h e method of f o r c e d R a y l e i g h s c a t t e r i n g i s t o b e m e n t i o n e d , s i n c e i t p r o v i d e s i n - f o r m a t i o n on s e l f - d i f f u s i o n , which i s n o t p o s s i b l e t o o b t a i n from o r d i n - a r y q u a s i e l a s t i c l i g h t s c a t t e r i n g on m o n o d i s p e r s e s y s t e m s C4,SJ. A l 1 t h e e x p e r i m e n t s m e n t i o n e d have t h e p a r t i c u l a r l y i n t e r e s t i n g a s p e c t t h a t t h e y y i e l d n o t o n l y r e s u l t s a b o u t t h e o r d i n a r y t r a n s p o r t c o e f f i c i e n t s b u t t h a t i t i s a l s o p o s s i b l e t o m e a s u r e t h e t i m e and w a v e v e c t o r depend- e n c e o f c e r t a i n c o r r e l a t i o n f u n c t i o n s d i r e c t l y . The c o r r e l a t i o n f u n c - t i o n s c o n t a i n more i n f o r m a t i o n a b o u t t h e dynamics of t h e s y s t e m u n d e r c o n s i d e r a t i o n t h a n t h e t r a n s p o r t c o e f f i c i e n t s .

The aim o f t h i s p a p e r i s t o r e v i e w r e c e n t r e s u l t s [ 6 ] a b o u t a u n i f i e d

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

e d on methods d e v e l o p e d f o r t h e dynamic o f s i m p l e 1 i q u i d s . T h e p r o p e r t i e s

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985317

of these suspensions can nearly quantitatively be understood by using liquid state theory; it is of course necessary to introduce some important changes which take into account the fact that the macro- ions of a suspension move in a resisting medium.

2. TRANSPORT EQUATIONS AND COLLECTIVE PROPERTIES

The simplest type of space and time dependent behavior of the particle concentration c(r,t) results from employing the continuity equation

and to write for the long-wave-length current fluctuations J(x,t) Fick's law

j(r,t)

- Dc c(r,t) ,

-

(2.2)

where Dc is the collective or mass-diffusion coefficient. The equa- tion (2.2) is however only a valid description in the hydrodynamic regime, where typical wavelengths are large compared to the average distance between particles. The interesting aspect of the experiments, which were mentioned in the introduction, is however the fact that in light scattering it is possibie to detect fluctuations of the system on the scale of several 1000 A, which is of the same order as the average distance between particles. Therefore, the system of inter- acting Brownian particles cannot be considered as a continuum. The scattering experiments probe the system on the length scale of its short-range structure and detect the temporal changes of this struc- ture.

It is therefore necessary to take into account that the particle cur- rent at (r,t) depends on concentration gradients at neighboring posi- tions r' and at earlier times t'< t. The corresponding generalization of (2.2) is

t

j(5t) = - / dt' /d3r' D(1-r',t-t') V' c(rl,t') ; t3O (2.3)

- - -

O

The function D(r,t), called the generalized diffusion function, re- places the hydrodynamic quantity Dc in this non-local formulation.

From (2.1) and (2.3) one can easily obtain the dynamic structure fact- or, which is defined by

S(5,t)

< c(k,t) c(-&,O)> - ^(2-4)

Here, c(&,t) is the Fourier transform of the concentration fluctu- ation c(~,t); the bracket indicates an equilibrium average. Intro- ducing Laplace transforms, eqs. (2.1) and (2.3) yield

where S(k) = S(k,t=O) is the static structure factor. If one would

have used (2.2)-instead of (2.3), the Laplace transform D(k,z) of the

generalized diffusion function D(r,t) wzuld simply be replaced by Dc,

showing that the hydrodynamic limit of D(&,z) is equal to the mass-

diffusion coefficient:

I n t h i s l i m i t t h e dynamic s t r u c t u r e f a c t o r i s a p u r e e x p o n e n t i a l i n t ime

lim S ( k , t ) = S(0) exp(- Dc i 2 t ) k 4 , t-

kzt=cons t

i t s d e c a y i s d e t e r m i n e d by t h e m a s s - d i f f u s i o n c o e f f i c i e n t D .

The e x t e n s i o n of ( 2 . 2 ) t o ( 2 -3) and a s a consequence o f it t h e i n t r o - d u c t i o n o f a wave-number and f r e q u e n c y - d e p e n d e n t d i f f u s i o n f u n c t i o n h a s h e r e been d e m o n s t r a t e d , s i n c e i t s e r v e s a s a n i l l u s t r a t i o n o f what ap- p e a r s i n a more s y s t e m a t i c t h e o r y , which i s now b e i n g o u t l i n e d [6] .The s t a r t i n g p o i n t of t h i s t h e o r y i s t h e Fokker-Planck e q u a t i o n f o 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 (pl , . . . ,EN , r l , . . . , r t ) 2 f (i',t) o f t h e momen-

-N '

t a p i a n d c o o r d i n a t e s r . o f t h e N i n t e r a c t i n g m a c r o p a r t i c l e s a t t i m e t :

-1

where t h e F o k k e r - P l a n c k o p e r a t o r i s g i v e n by

n = - l { l p ^a .- a 1 + . l . -.<. a ^. . ^{( k T - a + - 1 p . )} ( 2 . 9 ) rn -i'K

Li ap. api

B ap. m-J

i

,J -J

Here, Fi ^{i s} t h e f o r c e a c t i n g on p a r t i c l e i by a l 1 o t h e r m a c r o p a r t i c l e s , and 5 . - d e n o t e s t e n s o r s of hydrodynamic i n t e r a c t i o n . S i n c e i t i s w e l l

=l]

e s t a b l i s h e d t h a t hydrodynamic i n t e r a c t i o n s a r e u n i m p o r t a n t i n t h e h i g h l y c h a r g e d p o l y s t y r e n e s y s t e m s , t h e y w i l l be n e g l e c t e d i n what f o l l o w s , s o t h a t sii

^{50 6ii} I, ^{where 5'} i s t h e f r i c t i o n c o e f f i c i e n t

- - J d

-

of a n o n i n t e r a c t i n g m a c r o p a r t i c l e . I t s h o u l d be n o t e d t h a t t h e f i r s t term of ( 2 . 9 ) i s i d e n t i c a l t o t h e L i o u v i l l e o p e r a t o r , and t b a t t h e s e - cond t e r m . ~ r o ~ o r t i o n a l , . . t o t h e f r i c t i o n c o n s t a n t co,makes R non-her- m i t i a n .

The u s e o f t h e Fokker-Planck e q u a t i o n r e s t s upon t h e a s s u m p t i o n t h a t t h e system i s b e i n g i n v e s t i g a t e d e x p e r i m e n t a l l y on a t i m e s c a l e on which t h e s o l v e n t , t h e c o u n t e r - i o n s and p o s s i b l e i o n s a r e a l w a y s i n e q u i l i b r i u m w i t h t h e l a r g e Brownian p a r t i c l e s . I t i s assumed t h a t d e - v i a t i o n s o f t h e s m a l l m o l e c u l e s and i o n s from t h i s e q u i l i b r i u m r e l a x f a s t e r t h a n t h e Brownian r e l a x a t i o n t i m e

m / c 0 o f a s i n g l e macro- p a r t i c l e . T h i s t i m e i s o f t h e o r d e r of 10-8 s e c , w h e r e a s t h e s h o r t e s t c o r r e l a t i o n t i m e s i n q u a s i e l a s t i c l i g h t s c a t t e r i n g a r e o f t h e o r d e r o f 10-6 s e c and l o n g e r .

Having e s t a b l i s h e d a t r a n s p o r t e q u a t i o n f o r t h e system one c a n s t u d y t h e t i m e - d e p e n d e n t c o r r e l a t i o n f u n c t i o n s o f t h e b a s i c p h a s e - s p a c e v a r i a b l e s , which a r e 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

and t h e c u r r e n t f l u c t u a t i o n s

The t i m e d e r i v a t i v g s o f ? ( & ) _ a n d f(&) a r e g i v e n by o p e r a t i n g w i t h t h e h e r m i t i a n a d j o i n t R + , s i n c e R i s n o n - h e r m i t i a n . From ( 2 . 9 ) and ( 2 . 1 0 ) one o b t a i n s t h e c o n t i n u i t y e q u a t i o n f o r t h e p h a s e s p a c e v a r i a b l e s

and f o r t h e c u r r e n t f l u c t u a t i o n s one f i n d s

rn R+ - - j ( k )

-i - k a(%) ^{+ i ( k )} . ( 2 . 1 3 ) The f i r s t t e r m i s t h e g r a d i e n t o f t h e ( o s m o t i c ) p r e s s u r e ,

G ( k ) = k B ~ - C(k)/S(k), and t h e s e c o n d one r e p r e s e n t s a f l u c t u a t i n g

f o r c e d e n s i t y

f (5) = i - c0 .

- -

Here, a(&) ^{i s} t h e v i s c o u s s t r e s s t e n s o r of t h e " l i q u i d of i n t i r a c t i n g macroions" and i s f o r m a l l y i d e n t i c a l t o t h e e x p r e s s i o n known i n t h e t h e o r y o f s i m p l e l i q u i d s .

As c a n be s e e n from (2.12) t o ( 2 . 1 4 ) , t h e time d e r i v a t i v e s of -? ('() and

i(&) behave d i f f e r e n t l y a s k+O.Whereas 6+ c ( k )

O f o r k + O , R j(k) O i n t h i s l i m i t . T h i s e x p r e s s e s t h e f a c t t h a t - t h e t o t a l c o n c e n t r a t i o n i s a c o n s e r v e d q u a n t i t y of t h e system of Brownian p a r t i c l e s whereas t h e c u r r e n t o r t h e momentum i s n o t . Momentum and e n e r g y a r e c o n t i n u o u s l y exchanged between t h e subsystem o f i n t e r a c t i n g m a c r o p a r t i c l e s and t h e s o l v e n t . T h i s d i f f e r e n c e , compared t o t h e s i m p l e l i q u i d , a r i s e s from t h e second term i n (2.14) which i s 3 consequence of t h e second p a r t of

( 2 . 9 ) , t h e "Fokker-Planck p a r t " of R .

The p r i n c i p a l c o r r e l a t i o n f u n c t i o n s t o s t u d y a r e t h e dynamic s t r u c t u r e f a c t o r

s(k,t) = < ;(&) St ^;(-XI>

^, ( 2 . 1 5 ) and t h e l o n g i t u d i n a l and t r a n s v e r s e c u r r e n t c o r r e l a t i o n f u n c t i o n s

4 J,,

and TL d e n o t e t h e components of f(lc) p a r a l l e l and p e r p e n d i c u l a r t o k , r e s p e c t i v e l y . I n t h e f o l l o w i n g we c a n r e s t r i c t o u r s e l v e s t o t h e -

c a l c u l a t i o n of

t ) and CA (ic, t ) , s i n c e Cl, (k, t ) i s s i m p l y r e l a t e d t o S ( & , t ) b e c a u s e o f t h e c o n t i n u i t y e q u a t i o n .

A v e r y c o n v e n i e n t way t o d e r i v e e x p r e s s i o n s f o r t h e c o r r e l a t i o n f u n c - t i o n s from t h e t r a n s p o r t e q u a t i o n ( 2 . 8 ) and t h e e q u a t i o n s of motion

(2.12) and (2.13) f o r t h e v a r i a b l e s which appear i n (2.15) t o (2.171 ,

i s t h e Mori-Zwanzig p r o j e c t i o n o p e r a t o r t e c h n i q u e . S i n c e t h e d e r i - v a t i o n i s g i v e n e l s e w h e r e [6,7], we w i l l m e r e l y s t a t e t h e r e s u l t s and

t r y t o i l l u s t r a t e them i n p h y s i c a l t e r m s . By employing a p r o j e c t i o n o p e r a t o r which p r o j e c t s a l 1 phase s p a c e v a r i a b l e s o n t 0 t h e c o n c e n t r a - t i o n v a r i a b l e s , which a r e t h e slow v a r i a b l e s of Our s y s t e m , one ob- t a i n s f o r S ( k , z) , thedLaplace t r a n s f o r m o f (2.15) , an e q u a t i o n i d e n - t i c a l t o (2.5) , where D(&, z ) k 2 i s t h e memory f u n c t i o n of t h e c o n c e n t r a - t i o n a u t o c o r r e l a t i o n f u n c t i o n and where D(&,z) i s now g i v e n a s a c o r - r e l a t i o n f u n c t i o n of t h e l o n g i t u d i n a l c u r r e n t f l u c t u a t i o n s

The main d i f f e r e n c e of t h i s e x p r e s s i o n compared t o (2.16) i s t h e ap- p e a r a n c e o f Q , which p r o j e c t s on t h e s u b s p a c e o r t h o g o n a l t o c l & ) . I n c o n t r a s t t o tfie p u r e l y phenomenological i n t r o d u c t i o n o f t h e g e n e r a l i z - ed d i f f u s i o n f u n c t i o n i n e q . ( 2 . 3 ) , % ( & , z ) i s now g i v e n by a n e x p l i c i t e x p r e s s i o n .

A s a n e x t s t e p one c a n r e p e a t t h e p r o c e d u r e once more and w r i t e 2

memory ecluation f o r E ( k . z ) , employing now a p r o j e c t i o n o p e r a t o r P . which p r o j e c t s on x, ^(k). The r e s u l t c a n be w r i t t e n a s 1 '

which can be c o n s i d e r e d a s a g e n e r a l i z a t i o n of t h e S t o k e s - E i n s t e i n r e -

l a t i o n t o f i n i t e k and z . To s e e t h i s one t a k e s t h e hydrodynamic

l i m i t ( k , z

O ) i n (2.19) and s p e c i a l i z e s t o a n o n - i n t e r a c t i n g s y s t e m ,

f o r which S ( k )

1. Using ( 2 . 6 ) one o b t a i n s from (2.19)

which identifies,,: (0,O) w i t h t h e s i n g l e p a r t i c l e f r i c t i o n c o e f f i ~ i e n t

G O .

T h e r e f o r e , 3, ( k , z ) , which a p p e a r s a s t h e memory f u n c t i o n of D(&,z) i n e q . ( 2 . 1 9 ) , g e n e r a l i z e s t h e f r i c t i o n c o e f f i c i e n t 5' t o a k and z depen- d e n t l o n g i t u d i n a l f r i c t i o n f u n c t i o n . T h i s f u n c t i o n i s found t o c o n s i s t of two p a r t s :

-

^{k 2 -}

C,,(&,z)

5

7 n,,(k,z) . ( 2 . 2 1 )

The f i r s t one e x p r e s s e s t h e f a c t t h a t e v e r y p a r t i c l e e x p e r i e n c e f r i c t - - i o n , b e c a u s e of t h e p r e s e n c e of t h e s o l v e n t , w h e r e a s t h e s e c o n d t e r m a r i s e s from t h e dynamic l o n g i t u d i n a l v i s c o s i t y of t h e l i q u i d of i n t e r - a c t i n g Brownian p a r t i c l e s .

I f one u s e s a s i m i l a r p r o c e d u r e f o r t h e t r a n s v e r s e c u r r e n t c o r r e l a t i o n f u n c t i o n , one f i n d s

where n s @ , z ) i s t h e dynamic s h e a r v i s c o s i t y of t h e l i q u i d o f Brownian

p a r t i c l e s .

The v i s c o s i t y f u n c t i o n s

z) and

( k , z) a p p e a r i n g i n e q . ( 2 . 2 1 ) and ( 2 . 2 2 ) a r e found from t h e Mori-Zwanzig f o r m a l i s m t o b e g i v e n by

*

- - - -

,, ( k , z )

@ < o ^{( k ) [z - Q r;} QI-' o z z ( - k ) >

( 8 -

V ZZ

( 2 . 2 3 )

and an i d e n t i c a l e x p r e s s i o n f o r ( k , z ) e x p e c t t h a t t h e i n d i c e s on

/r

a ( + k ) a r e z x i n s t e a d of z z . ~ e r e k h a s b e e n c h o s e n i n t h e z d i r e c t i o n o f t h e c o o r d i n a t e s y s t e m . I t s h o u l d b e n o t e d t h a t t h e e x p r e s s i o n s f o r

;i,, ( k , z ) a n d Ti ( k , z ) , a r e f o r m a l l y i d e n t i c a l t o t h o s e f o r a s i m p l e l i q u i d [8JT The o p e r a t o r Q i n (2.23) p r o j e c t s on t h e s u b s p a c e o r t h o g o n a l t o E ('0 and i ( k ) .

So f a r we have r e d u c e d t h e c o r r e l a t i o n f u n c t i o n S ( k , t ) t h r o u g h e q s . ( 2 . 5 ) , (2.19) and ( 2 . 2 1 ) t o G,, ( k , z ) s o t h a t t h e dynamic s t r u c t u r e f a c t o r i s _- g i v e n by

S ( k )

S ( k , z ) -

; ~ ( k )

( 2 . 2 4 )

2 2

T -

k

(k) ^m S ( k )

Z f

+ ( C O +

a,ci,~ii

and C , ( k , t ) , e q . ( 2 . 2 2 ) , i s r e d u c e d t o ; & k , z ) . These v i s c o s i t y f u n c t i o n s have been c a l c u l a t e d i n mode-coupling a p p r o x i m a t i o n ; t h e r e s u l t i s 161

where g a r e t h e components o f

Z I

x

i kgT g ( k , k ' )

-

; O( h ( k ; )

5; h(k;))

2 S ( k ; ) S ( k l ) -

-2

and ki ^,2

^(1/2) k kt . S i n c e h(&)

(S(k) - 1 ) / c i s t h e F o u r i e r t r a n s f o r m e d t o t a l c o r r e l a t i o n f u n c t i o n , t h e v i s c o s i t y f u n c t i o n s i n

( 2 . 2 5 ) , (2.26) a r e e n t i r e l y e x p r e s s e d by t h e s t a t i c s t r u c t u r e f a c t o r S ( k ) and t h e dynamic s t r u c t u r e f a c t o r S ( k , t ) . I n t h i s way, a s e l f - c o n s i s t e n t s e t of c o u p l e d e q u a t i o n s i s o b t a i n e d , which c a n be s o l v e d n u m e r i c a l l y . A s an i n p u t one u s e s t h e s t a t i c s t r u c t u r e f a c t o r S ( k ) , which c a n e i t h e r b e t a k e n from e x p e r i m e n t o r c a n be c a l c u l a t e d from a one-component macrof l u i d mode1 , a s developed by Hansen and H a y t e r 193

f o r t h e h i g h l y c h a r g e d d i l u t e s u s p e n s i o n s . I t s h o u l d b e n o t e d t h a t t h e r e i s no a d j u s t a b l e p a r a m e t e r i n t h e dynamical t h e o r y .

Before t h e r e s u l t s of t h i s t h e o r y a r e r e l a t e d t o e r p y r i m e n t , one s i m p l i - f i c a t i o n can be i n t r o d u c e d . I n eq. (2.24) cO/m

i s o f t h e o r d e r of 108 s e c - l . On t h e o t h e r hand, t h e q u a s i e l a s t i c l i g h t s c a t t e r i n g method c a n o n l y r e s o l v e t i m e s l a r g e r t h a n a b o u t 1 0 - b s e c , s o t h a t t h e v a r i a b l e z i n (2.24) i s s m a l l e r t h a n 106. T h e r e f o r e , (2.24) s i m p l i f i e s

We w i l l now d i s c u s s some e x p e r i m e n t s i n which t h e v i s c o s i t y f u n c t i o n s a p p e a r . A s a f i r s t example, we mention t h e second cumulant of S ( k , t ) . One expands S ( k , t ) f o r s h o r t t i m e s a s

From (2.28) t h e f i r s t two cumulants a r e g i v e n by

,, ^{(k) = - ,Jl(k)} k q,,(k,~)/(c

c0 1.

2

-

I t i s s e e n t h a t t h e i n i t i a l v a l u e of n , , ( k , t ) d e t e r m i n e s t h e second cumulant (k) . T h i s q u a n t i t y i s , however, d i f f i c u l t t o d e t e r m i n e e x p e r i m e n t a S l ~ w i t h s u f f i c i e n t a c c u r a c y . I t i s more c o n v e n i e n t t o d e t e r m i n e a mean r e l a x a t i o n time f o r S ( k , t ) . Using (2.28) one f i n d s

The mean r e l a x a t i o n of t h e dynamic s t r u c t u r e f a c t o r is t h e r e f o r e de- t e r m i n e d by t h e k-dependent l o n g i t u d i n a l v i s c o s i t y q,,(k), s i n c e

m

:(L,O) =

/ d t

:

.

O

(2.33)

From t h e dynamic l i g h t s c a t t e r i n g e x p e r i m e n t one c a n d e t e r m i n e t h e q u a n t i t y

P l ( L )

-

b ( L ) E

(2.34)

P , ( 5 )

which from (2.30) and (2.32) can be e x p r e s s e d a s . -

1

!i7 n , , ( ! g ( c c P )

S i n c e S k ) i s j u s t t h e time i n t e g r a l of t h e mode c o u p l i n g r e s u l t ( 2 . 2 5 ) , one can Compare t h e o r y and e x p e r i m e n t . Using t h e method developed by Hansen and H a y t e r [9] t o c a l c u l a t e t h e s t a t i c s t r u c t u r e f a c t o r f o r t h e s y s t e m s s t u d i e d e x p e r i m e n t a l l y by Grüner and Lehmann 1101, good a g r e e - ment was o b t a i n e d f o r A(&), which measures t h e d e v i a t i o n of S ( & , t ) from a p u r e e x p o n e n t i a l f u n c t i o n of time [ 6 ] . I t i s found t h a t A(&) i s l a r g e s t a t around 2km/3, where km i s t h e p o s i t i o n of t h e f i r s t maxi- mum of S (&) .

An e s p e c i a l l y i n t e r e s t i n g f e a t u r e of t h e r e s u l t s of Grüner and Lehmann was t h e a p p a r e n t c o n c e n t r a t i o n independence of t h e e x p e r i m e n t a l A(&),

i f i t was p l o t t e d a s a f u n c t i o n o f k/km. I n f a c t , Our t h e o r e t i c a l c a l - c u l a t i o n s show o n l y a v e r y weak c o n c e n t r a t i o n dependence i n t h e con- c e n t r a t i o n regime where t h e e x p e r i m e n t s were performed. An i n s i g h t i n t o t h e c o n c e n t r a t i o n dependence o f t h e v i s c o e l a s t i c e f f e c t i s ob- t a i n e d by l o t t i n g t h e maximum o f A(&) a s a f u n c t i o n o f volume concen- t r a t i o n [6! . I t i s found t h a t t h i s maximum i n c r e a s e s s t r o n g l y a t low c o n c e n t r a t i o n s and i s o n l y weakJy i n c r e a s i n g i n t h e r e g i o n of con- c e n t r a t i o n s i n v e s t i g a t e d i n ~ e f . [ ~ o J .

From t h i s d i s c u s s i o n i t becomes c l e a r why t h e measured c o r r e l a t i o n f u n c - t i o n s i n q u a s i - e l a s t i c l i g h t s c a t t e r i n g a r e n o n - e x p o n e n t i a l a t f i n i t e s c a t t e r i n g v e c t o r s . The 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 , which g i v e r i s e t o l i g h t s c a t t e r i n g , a r e coupled t o s t r e s s s f l u c t u a t i o n s of t h e h i g h - l y c o r r e l a t e d l i q u i d of i n t e r a c t i n g c o l l o i d a l p a r t i c l e s . The s t r e s s f l u c t u a t i o n s a r e d e t e r m i n e d by t h e n o n - t r i v i a l v i s c o e l a s t i c b e h a v i o r of t h e s y s t e m , a s d e s c r i b e d by t h e f u l l v i s c o s i t y f u n c t i o n s q , , , s ( & , t ) . I f S ( & , t ) would be a p u r e e x p o n e n t i a l , t h e t i m e ~ ( & ) d e f i n e d i n (2.32) would be e q u a l t o t h e i n v e r s e of t h e f i r s t cumulant and a s a conse- quence A(&) would v a n i s h . S i n c e t h i s i s n o t t h e c a s e , A ( & ) i s a d i r e c t measure of n,, (&) , a c c o r d i n g t o (2.34a) .

As shown i n c o n n e c t i o n w i t h e q . ( 2 . 3 1 ) , t h e i n i t i a l v a l u e n,,(&,O) of q , , ( & , t ) d e t e r m i n e s t h e s h o r t - t i m e b e h a v i o r of S ( & , t ) . I t i s e x p e c t e d t h a t t h e i n i t i a l v a l u e s of t h e v i s c o s i t y f u n c t i o n s w i l l i n d e e d de- t e r m i n e o t h e r p r o p e r t i e s of t h e system a t s h o r t t i m e s . One c a n show

[ 6 ] t h a t t h e m a c r o s c o p i c c u r r e n t f l u c t u a t i o n s a t s h o r t t i m e s a r e governed by t h e f o l l o w i n g e q u a t i o n s of motion

T h e r e f o r e , t h e s y s t e n of i n t e r a c t i n g c o l l o i d a l p a r t i c l e s behaves on a

s h o r t time s c a l e l i k e a damped harmonic o s c i l l a t o r . A t s h o r t t i m e s

t h e system h a s r e s t o r i n g f o r c e s and shows e l a s t i c b e h a v i o r . As i n t h e

t h e o r y o f s i m p l e l i q u i d s one can t h e r e f o r e i n t r o d u c e h i g h - f r e q u e n c y

e l a s t i c c o n s t a n t s a s a measure of t h e r e s t o r i n g f o r c e s :

A s i n i t i a l v a l u e s o f t i m e - d e p e n d e n t c o r r e l a t i o n f u n c t i o n s t h e y c a n b e e x p r e s s e d by e q u i l i b r i u m q u a n t i t i e s :

3 4 ~ " ( k ) + ~ ~ ( k ) = c 3 kBT + c jd3r g ( r ) a 2 u ( r ) 1 - ^cos k z

a z 2 k 2 ] ( 2 . 3 9 )

B ( k ) = c [LB~ + c jd3r g ( r ) - ( 2 . 4 0 )

a x 2 k 2

Only 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 g ( r ) a n d t h e cwo p a r t i c l e i n t e r - a c t i o n p o t e n t i a l U ( r ) d e t e r m i n e t h e h i g h - f r e q u e n c y e l a s t i c c o n s t a n t s . F o r t h e s y s t e m o f G r ü n e r a n d Lehman 1101 t h e r e s u l t s 171 f o r t h e k- d e p e n d e n t h i g h - f r e q u e n c y e l a s t i c c o n s t a n t s f o r 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 shown i n P i g . 1 . I t i s s e e n t h a t t h e l o n g i t u d i n a l modulus E ( k ) = ( 4 / 3 ) G a ( k ) + K" ( k ) a t s m a l l k i s more t h a n one o r d e r o f m a g n i t u d e l a r g e r t h a n t h e s h e a r modulus Ga ( k ) . The v a l u e s o f k a t w h i c h t h e two m o d u l i a p p r o a c h e a c h o t h e r i s a b o u t e q u a l t o t h e r e c i - p r o c a l mean d i s t a n c e between n e i g h b o r i n g p a r t i c l e s . The c o n c e n t r a - t i o n d e p e n d e n c e of t h e 10 - w a v e l e n g t h l i m i t o f t h e m o d u l i i s f o u n d t o b e E ( 0 ) - c 2 a n d G (O)- c l y p . The q u a d r a t i c d e p e n d e n c e o n c o n c e n t r a - t i o n o f t h e l o n g i t u d i n a l modulus was f o u n d e x p e r i m e n t a l l y b y G r ü n e r and Lehmann [ll] from t h e i r l i g h t s c a t t e r i n g d a t a .

F i g u r e 1. H i g h - f r e q u e n c y e l a s t i c c o n s t a n t s a s a f u n c t i o n of kais=ak4( 1 / 3 f o r f o u r volume f r a c t i o n s

0.5; 1 . 5 ; 3 . 0 a n d 7 . 5 x 1 0 - .

F u l l l i n e s r e p r e s e n t t h e l o n g i t u d i n a l modulus 4

a 3 ~ ( k ) / ( 3 kgT) ; d a s h e d l i n e s t h e s h e a r modulus 4 a a 3 Gm ( k ) / ( 3 kgT) .

a i s t h e r a d i u s of t h e c o l l o i d a l p a r t i c l e s .

From e q s . ( 2 . 3 1 ) , ( 2 . 3 7 ) and ( 2 . 3 9 ) i t i s c l e a r t h a t a n a c c u r a t e d e - t e r m i n a t i o n o f t h e s e c o n d c u m u l a n t o f S ( & , t ) would y i e l d t h e &-de- p e n d e n c e o f t h e l o n g i t u d i n a l modulus E ( & ) . S i n c e t h e l a t t e r i s d i r e c t - l y r e l a t e d t o t h e p a i r p o t e n t i a l , e q . ( 2 . 3 9 ) , t h i s c o u l d l e a d t o a d e t e r m i n a t i o n o f t h e F o u r i e r components U(k) o f t h i s p o t e n t i a l . T h i s i s s e e n by r e w r i t i n g ( 2 . 3 9 ) a s

2

- - - Ï

k k " ( k ) - [ L - ( ~ - l y ) ] U(-Ca),}

H e r e , h ( k ) i s t h e F o u r i e r t r a n s f o r m o f h ( r ) = g ( r ) - 1. The n u m e r i c a l

e v a l u a t i o n shows, t h a t f o r s t r o n g l y c h a r g e d p o l y s t y r e n e s p h e r e s and

s u f f i c i e n t l y s m a l l w a v e v e c t o r s o n l y t h e f i r s t t e r m i n ( 2 . 4 1 ) i s o f

importance. T h e r e f o r e , t o a good a p p r o x i m a t i o n

lJ2(k)

- -

( k )

- 6 2 U(k) -

1

-

u,(k) c

As a f i n a l comment a b o u t t h e a p p e a r a n c e of t h e v i s c o s i t y f u n c t i o n s a r e l a t i o n between t h e s t a t i c s h e a r v i s c o s i t y

50 (k=O) and t h e s h e a r modulus i s m e n t i ~ n e d . ~ The mode c o u p l i n g r e s u ? t 72.26) shows [6] numer-

i c a l l y t h a t O,t)/G ( O ) i s o n l y a f u n c t i o n of ~ O t / ( 2 a ~ ~ ) ~ , where a

( 3 / 4 r c j i j 3 i s t h e i o n - s p h e r e r a d i u s . T h e r e f o r e

i s

m 03

: j d t n S ( o , t )

~ ~ ( 0 ) /dt f(D0t/(2aiS)

)

O O

2

(2.43)

V a i s ) "

= ~ ~ ( 0 ) ---- ^JdT

DO O

(2ais/10)

The i n t e g r a l i s of o r d e r 1 0 - ~ . T h e r e f o r e qS/Grn(o)

DO

S i n c e 2ais i s of t h e same t h e mean i n t e r p a r t i c l e d i s t a n c e , t h i s r e l a t i o n i s v e r y s i m i l a r r e s u l t , which was o b t a i n e d e m p i r i c a l - l y by C h a i k i n and coworkers by measuring t h e e l a s t i c s h e a r mo- d u l u s i n t h e c r y s t a l l i n e phase and

i n t h e f l u i d p h a s e .

3. SINGLE-PARTICLE PROPERTIES

S i n g l e - p a r t i c l e p r o p e r t i e s of c o r r e l a t e d s y s t e m s a r e c o n c e p t u a l l y simp- l e r t o u n d e r s t a n d t h a n c o l l e c t i v e o n e s ; t h e y l e a d i n a r a t h e r d i r e c t way t o e f f e c t s which d e s c r i b e t h e h i g h l y c o r r e l a t e d s h o r t - r a n g e s t r u c t - u r e and i t s r e l a x a t i o n . On t h e o t h e r hand, t h e r e a r e l e s s e x p e r i m e n t - a l r e s u l t s a b o u t s i n g l e - p a r t i c l e p r o p e r t i e s , a l t h o u g h r e c e n t measure- ments of t h e s e l f - d i f f u s i o n c o n s t a n t Ds u s i n g t h e f o ~ c e d R a y l e i g h s c a t -

t e r i n g t e c h n i q u e have r e s u l t e d i n a f i r s t s y s t e m a t i c i n v e s t i g a t i o n of t h e dependence of Ds on volume c o n c e n t r a t i o n and S a l t [5,12].

For 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 o f s i n g l e - p a r t i c l e p r o p e r t i e s one c a l c u - l a t e s t h e p r o b a b i l i t y G ( r , t ) t h a t a tagged p a r t i c l e (denoted a s p a r t - i c l e 1 ) , ^{which a:} (O) a t rime O , h a s i t s p o s i t i o n a t f l ( t ) a t t h e i a t e r t i m e t . The E o u r i e r t r a n s f o r m o f G ( r , t ) i s

which c a n be c a l c u l a t e d a l o n g s i m i l a r l i n e s a s S ( & , t ) and C,(k,t) i n t h e p r e v i o u s s e c t i o n . The r e s u l t i s r6]

," , ..

G(k,z) - ⁼ cz

D*(k,z) k 1 (3.2)

where t h e g e n e r a l i z e d s e l f - d i f f u s i o n f u n c t i o n

Y

i s g i v e n by

( k , z ) = c0

A ~ ~ ( k , z ) -

S

-

( 3 . 3 ) w i t h t h e f o l l o w i n g mode-coupling e x p r e s s i o n f o r A<,(&,t)

Here c ( k )

[ ~ ( k ) - 11 ^/ s(&)] i s t h e d i r e c t c o r r e l a t i o n f u n c t i o n .

The e q B a t i o n s (3.2) t o ( 3 . 4 ) t o g e t h e r w i t h t h e r e s u l t s f o r S ( & , t ) o f

t h e p r e v i o u s s e c t i o n form a g a i n a c l o s e d s e t of e q u a t i o n s , which r e -

duce a l 1 s i n g l e - p a r t i c l e p r o p e r t i e s t o t h e s t a t i c s t r u c t u r e f a c t o r , which d e t e r m i n e s t h e k e r n e l i n ( 3 . 4 ) .

W r i t i n g as ^{@, z)}

A$(&, z) , t h e mean s q u a r e d i s p l a c e m e n t i s g i v e n

by 1 t

~ ( t ) < b i ( t ) - Il(o)j2>= DO^ - 1 d t t ( t - t t ) A D ( o , ~ * ) . ( 3 . 5 )

O

Thus, W(t)

t a t s h o r t t i m e s , whereas W(t)+Dst f o r t-+- which de- f i n e s t h 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 Ds. Fig.2 shows Our r e s u l t s f o r f o u r d i f f e r e n t c o n c e n t r a t i o n s i n comparison t o t h e c a s e o f f r e e d i f - f u s i o n . A t s h o r t t i m e s t h e p a r t i c l e s t a r t s t o d i f f u s e a s a f r e e p a r t - i c l e , l a t e r t h e r e p u l s i v e i n t e r a c t i o n s from n e i g h b o r i n g p a r t i c l e s a r e f e l t , which r e s u l t s i n a slowing down o f t h e growth of W ( t ) . I t i s a l s o e v i d e n t from Fig.2 t h a t i n t h e more c o n c e n t r a t e d systems t h e d e v i a t i o n from t h e i n i t i a l s l o p e s e t s i n a t e a r l i e r t i m e s .

F i g u r e 2. Mean-square d i s p l a c e m e n t o f a c o l l o i d a l p a r t i c l e a s a f u n c t i o n of t h e reduced time

~ 0 t / ( 2 a . , ) ~ f o r f o u r d i f f e r e n t volume f r a c t i o n s between 0.2 ^{x 10-3}

and 7 . 5 x 10-3. The broken l i n e c o r r e s p o n d s t o f r e e d i f f u s i o n .

Another q u a n t i t y o f i n t e r e s t i s t h e v e l o c i t y a u t o c o r r e l a t i o n f u n c t i o n

I t i s found t h a t AD(0,t) 30 and t h a t AD(0,t) d e c a y s f a s t e r w i t h i n - c r e a s i n g c o n c e n t r a t i o n . T h i s d e c a y i s n o t a s i m p l e e x p o n e n t i a l . A t v e r y long t i m e s , V ( t ) h a s an a l g e b r a i c long-time t a i l , V ( t ) - t - 5 / 2 , which i s c h a r a c t e r i s t i c o f an overdamped system.

The s e l f - d i f f u s i o n c o e f f i c i e n t i s D

5 s ( 0 , 0 ) and i s shown i n F i g . 3 f o r t h e system p a r a m e t e r s c h a r a c t e r i z i n g t h e s u s p e n s i o n s i n v e s t i g a t e d

by Grüner and Lehmann [10]. A t s m a l l volume f r a c t i o n s Ds d e c r e a s e s

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

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

e r c o n c e n t r a t i o n s t h 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 v a r i e s much weaker

with concentration, since now the colloid particles themselves will also contribute to the screening, which leads to a kind of saturation.

If this interpretation is correct, the crossover concentration should increase with added salt. A detailed comparison of the predictions of this theory with the experimental results presented by B. _{Dozier et} al. at this Workshop Cl21 have still to be made.

Figure 3. Self-diffusion coefficient as function of volume fraction for a s stem of polystyrene spheres having diameter of 900 )i , surfa e potential 73 meV and a screening length of 5000 1.

4. CONCLUSION

The aim of this paper has been to demonstrate that the dynamical pro- perties of colloidal suspensions can be described and understood by using methods which can be called generalized hydrodynamics of an over- damped liquid. For those cases, where a direct comparison with ex- periments on highly charged spherical particles is possible, very good agreement between theory and experiment is obtained. This statement is of particular importance if one takes into account that the theory of the dynamical properties contains no adjustable parameter. The only input is the static structure factor, which can either be taken from static experiments on the same samples or from theoretical models like the one-component macrofluid mode1 of Hansen and Hayter [9]. The main result of the theory is that the highly correlated col.loida1 suspen- sion can be regarded as an overdamped liquid, which has local visco- elastic stress fluctuations due to the short-range order. These stress fluctuations lead to a slowing down of the relaxation of concentration fluctuations (as seen in the non-exponential behavior of S(&,t) ) and of the particle self-diffusion (as is evident from the mean-square displacement). The result as described in this paper were obtained by neglecting hydrodynamic interactions. Corresponding general expres- sions which include hydrodynamic interactions were given elsewhere[6].

For systems which contain rather large amounts of salt it will be

important io include hydrodynamic effects.

REFERENCES

1. Pusey, P.N. and R.J. A. Tough, in: Dynamic Light Scattering and Velocimetry: Applications of Photon Correlation Spectroscopy, edited by P. Pecora (New York: Plenum) 1982

2. See for instance, Hayter, J.B. and J. Penfold, J.Chem.Soc.Faraday Trans.l,Z, 1851 (1981); Kalus, J., H. Hoffmann, K. Reizlein, W. Ulbricht, and K. Ibel, Ber. Bunsenges, Phys.Chem.E, 37 (19823 3. Cebula, D.J., J. W. Goodwin, G.C. Jeffrey, R.H. Ottewill, A.

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