RHEOLOGY OF CONCENTRATED DISPERSIONS

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Submitted on 1 Jan 1986

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RHEOLOGY OF CONCENTRATED DISPERSIONS

R. Blanc

To cite this version:

R. Blanc. RHEOLOGY OF CONCENTRATED DISPERSIONS. Journal de Physique Colloques, 1986, 47 (C1), pp.C1-65-C1-71. �10.1051/jphyscol:1986110�. �jpa-00225494�

RHEOLOGY OF CONCENTRATED DISPERSIONS

R . BLANC

Departement de Physique des Systemes Desordonnes, Universite de Provence, U A 857 du C.N.R.S., F-13397 ~ a r s e i l l e Cedex 13, France

RESUME- C e t a r t i c l e p r e s e n t e d e maniere s u c c i n c t e l e s p r i n c i p a l e s g r a n d e u r s physiques q u i d o i v e n t etre p r i s e s en c o n s i d e r a t i o n d a n s l a rheologie d e s s u s p e n s i o n s c o n c e n t r e e s . On donne e n s u i t e les grandes l i g n e s d e s probl&nes a c t u e l s q u i se posent dans l ' e t u q e d e l a v i s c o s i t e d e s suspensions, l e u r v i t e s s e d e s e d i m e n t a t i o n e t l e s p r o c e s s u s d ' a g r e g a t i o n . On met en p a r t i c u l i e r e n a v a n t l e r8le joue par l ' e x i s t e n c e d e s t r u c t u r e s ordonnees ou a l g a t o i r e s .

ABSTRACT-In t h i s p a p e r we b r i e f l y p r e s e n t t h e main r e l e v a n t p h y s i c a l q u a n t i t i e s i n t h e rheology of c o n c e n t r a t e d d i s p e r s i o n s . Main l i n e s o f t h r e e t o p i c s ( s h e a r v i s c o s i t y , s e d i m e n t a t i o n v e l o c i t y and a g g r e g a t i o n p r o c e s s e s ) a r e t h e n g i v e n . The importance o r o r d e r e d or random s t r u c t u r e s is emphasized.

Nature and i n d u s t i a l p r o c e s s e s p r o v i d e numerous examples o f complex systems i n which s o l i d p a r t i c l e s a r e d i s p e r s e d i n a f l u i d : slip used i n t h e manufacture o f ceramics, n a t u r a l muds i n r i v e r s o r e s t u a r i e s , d r i l l i n g and mining f l u i d s , c e l l u l o s i c fibers i n paper pulp, p a s t r i e s , blood, c o n c r e t e and cements

...

Such systems a f f o r d v e r y v a r i o u s b e h a v i o u r s ; t h e i r mechanical, p h y s i c a l o r physicochemical p r o p e r t i e s are, a t t h e p r e s e n t t i m e , w i d e l y misunderstood. Such a l a c k o f understanding is due t o t h e complexity of t h e s e systems cumulating t h e d i f f i c u l t i e s o f random heterogeneous media and t h a t o f hydrodynamic systems. Due allowance b e i n g made f o r t h e i r eminent importance i n t h e i n d u s t r i a l o r n a t u r a l p r o c e s s e s , t h e y have been t h e s u b j e c t of numerous p r a c t i c a l s t u d i e s and a c o n s i d e r a b l e knowledge h a s been cumulated. But t h e r e is n o t , a t t h e p r e s e n t t i m e , a complete t h e o r e t i c a l corpus a l l o w i n g a deep understanding o f t h e s e systems.

When one c o n s i d e r s t h e c a s e o f one i s o l a t e d p a r t i c l e i n a f l u i d , some Simple s i t u a t i o n s as, f o r i n s t a n c e , t h e s e d i m e n t a t i o n i n a q u i e s c e n t f l u i d o r t h e behaviour o f t h e p a r t i c l e i n a s h e a r e d f l u i d may be e x a c t l y ( o r w i t h a v e r y good approximation) s o l v e d i f :

i ) t h e f l u i d is unbounded

i i ) t h e v i s c o u s f o r c e s p r e v a i l o v e r t h e i n e r t l a l o n e s ( l o w Reynolds' number regime) i i i ) t h e p a r t i c l e ' s shape is simple enough ( s p h e r e /I/, c y l i n d e r /2/, e l l i p s o i d

/3/ )

.

As soon as one o f t h e s e t h r e e c o n d i t i o n s is n o t f u l f i l l e d one d o e s not know t h e s o l u t i o n e x c e p t i n few s i t u a t i o n s . Among unsolved problems, some concern a s i n g l e s p h e r e e i t h e r i n f a s t s e d i m e n t a t i o n i n a n unbounded f l u i d /4/ o r i n slow s e d i m e n t a t i o n i n s i d e a c y l i n d r i c a l t u b e / 5 / . So it is not very s u r p r i s i n g t h a t more complex s i t u a t i o n s and e s p e c i a l l y t h o s e r e l a t e d t o c o n c e n t r a t e d d i s p e r s i o n s have n o t y e t been s o l v e d . I propose i n t h i s paper t o isolate the main p a r a m e t e r s r e l e v a n t t o t h e p r o p e r t i e s o f t h e s e systems and t o p u t forward some a s p e c t s o f t h e rheology o f d i s p e r s i o n s c l o s e l y r e l a t e d t o t h e s c i e n c e o f ceramics.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986110

J O U R N A L DE PHYSIQUE

I- M A I N PARAMETERS

Let u s f i r s t c o n s i d e r , i n a c o n t a i n e r , a d i s p e r s i o n o f i d e n t i c a l s p h e r e s i n a f l u i d . We suppose t h a t , due t o a s u i t a b l e s t i r r i n g , t h e d i s p e r s i o n is homogeneous : e x c e p t t h e s t a t i s t i c a l f l u c t u a t i o n s t h e number o f p a r t i c l e s by u n i t volume is t h e same everywhere i n t h e system. Let u s a p p l y t o t h i s two phase medium an e x t e r n a l f i e l d as, f o r i n s t a n c e , g r a v i t y . Then, t h e s p h e r e s go down o r up depending on t h e i r buoyancy- c o r r e c t e d weight. The p a t h followed by a g i v e n p a r t i c l e depends on t h e n e t r e s u l t o f f o r c e s and t o r q u e s e x e r t e d on it by t h e f l u i d , t h e e x t e r n a l f i e l d and t h e o t h e r p a r t i c l e s .

a p a r t o f t h e f o r c e and t o r q u e e x e r t e d by t h e f l u i d is due t o its molecular n a t u r e and corresponds t o t h e t r a n s l a t i o n n a l and r o t a t i o n n a l brownian motions. Another part is due t o t h e c o n v e c t i v e motion o f t h e f l u i d regarded as a continuous medium. The d e t e r m i n i s t i c s t r e s s e s e x e r t e d on t h e p a r t i c l e r e s u l t from t h e flow o f t h e f l u i d due t o t h e motion o f t h a t p a r t i c l e and o f a l l t h e o t h e r s . Each p a r t i c l e sediments i n a f l u i d p e r t u r k e d by t h e f a l l o f t h e o t h e r s . T h i s i n d i r e c t i n t e r a c t i o n between s p h e r e s s t r o n g l y depends on t h e volume f r a c t i o n i n t h e d i s p e r s i o n . I t is c a l l e d t h e hydrodynamic i n t e r a c t i o n . The r e l a t i v e importance o f t h e brownian and c o n v e c t i v e e f f e c t s ( P e c l e t ' s number), e s s e n t i a l l y depends on t h e size o f t h e p a r t i c l e . I n o r d i n a r y c o n d i t i o n s , t h e P e c l e t ' s number i s about one f o r s p h e r e s when t h e r a d i u s i s approximately e q u a l t o one micron. Brownian motion can, i n g e n e r a l , be h e g l e c t e d f o r p a r t i c l e s large^ t h a n few microns ( l a r g e P e c l e t ' s number regime).

When t h e s i z e o f t h e suspended p a r t i c l e s is i n t h e c o l l o i d a l range (1-1000m) t h e r e is a c o m p e t i t i o n between t h e s e d i m e n t a t i o n o f t h e p a r t i c l e and t h e i r brownian d i f f u s i o n : a c o n c e n t r a t i o n p r o f i l e t a k e s p l a c e i n t h e d i s p e r s i o n .

Such a c o n c e n t r a t i o n p r o f i l e is s t a b l e u n l e s s a n a g g r e g a t i o n p r o c e s s t a k e s p l a c e i n t h e suspension : i f two p a r t i c l e s can come very c l o s e each o t h e r due t o t h e i r brownian motion o r t h e flow o f t h e suspending f l u i d , t h e y can a g g r e g a t e under t h e a c t i o n o f Ver- W a a l s i n t e r a c t i o n /6/ s o t h a t t h e s i z e o f t h e " p a r t i c l e s " i n c r e a s e s and t h e d i s p e r s i o n f l o c c u l a t e s . Very o f t e n , t h e p a r t i c l e s g e t a s u r f a c e c h a r g e i n t h e f l u i d due t o an unequal d i s t r i b u t i o n of c h a r g e between its s u r f a c e and t h e f l u i d . Then, a c l o u d o f i o n s , c a l l e d t h e double-layer, s u r r o u n d s t h e p a r t i c l e and s c r e e n s its charge. The r e p u l s i v e f o r c e s between two p a r t i c l e s are reduced when t h e y a r e far from each o t h e r , b u t when hhey a r e c l o s e t h e o v e r l a p o f t h e c l o u d s p r e s e n t s t h e formation o f a g g r e g a t e s and s t a b i l i z e s t h e d i s p e r s i o n . N e v e r t h e l e s s t h e a d d i t i o n o f an e l e c t r o l y t e which i n c r e a s e s t h e i o n i c c o n c e n t r a t i o n a d reduces t h e e x t e n t o f t h e c l o u d may g r e a t l y modify t h e r e p u l s i v e s f o r c e s l e a d i n g t o an a g g r e g a t i o n o f p a r t i c l e s . So, an important parameter t o t a k e i n t o account is t h e i o n i c c o n c e n t r a t i o n i n t h e f l u i d and more g e n e r a l l y t h e physico-chemical p r o p e r t i e s o f t h e f l u i d , r e a s o n why one c a n add polymer or o t h e r i n g r e d i e n t s i n t h e system i n o r d e r t o a v o i d o r t o assist t h e f l o c c u l a t i o n .

L e t u s suppose now t h a t t h e weight o f t h e p a r t i c l e s is e x a c t l y compensated by t h e buoyancy f o r c e . I f o n e %poses known stresses o r known motions t o t h e d i f f e r e n t parts o f t h e A boundaries d e l i m i t i n g t h e d i s p e r s i o n a v e l o c i t y f i e l d and a s t r e s s f i e l d t a k e p l a c e i n t h e system. C l e a r l y , t h e behaviour o f t h e d i s p e r s i o n does n o t depend o n l y on t h e n a t u r e of t h e p a r t i c l e s and t h e f l u i d on t h e i n t e r a c t i o n s between p a r t i c l e s , b u t a l s o on t h e n a t u r e o f t h e s t r e s s e s o r t h e imposedmotion

.

- --

is e q u a l t o 1/2 G , where G i s t h e s h e a r r a t e . I n t h e same v e l o c i t y f i e l d , an e l l i p s o i d a l p a r t i c l e r o t a t e s w l t h a non c o n s t a n t a n g u l a r v e l o c i t y / 3 / . I f t h e p a r t i c l e is p r o l a t e ( c i g a r - l i k e ) it spends more time w i t h i t s a x i s o f r e v o l u t i o n p a r a l l e l t o t h e stream l i n e s t h a n p e r p e n d i c u l a r . The v i s c o s i t y o f d i s p e r s i o n o f p r o l a t e e l l i p s o i d s is g r e a t e r t h a n t h a t o f a d i s p e r s i o n o f s p h e r e s w i t h t h e same volume f r a c t i o n .

11- VISCOSITY OF CONCENTRATED DISPERSIONS

I n t h e f i r s t y e a r s o f t h i s c e n t u r y , E i n s t e i n , i n h i s work on brownian movment, computed t h e v i s c o s i t y ⁷⁾ o f a d i l u t e d i s p e r s i o n o f s p h e r e s and o b t a i n e d t h e w e l l known r e l a t i o n :

where qo is t h e v i s c o s i t y o f t h e suspending f l u i d and @ is t h e volume f r a c t i o n /7/.

T h i s law is c o r r e c t o n l y when t h e volume f r a c t i o n i s very ^weak : a few p e r c e n t / a / . Due t o hydrodynamic i n t e r a c t i o n between s p h e r e s n o t t a k e i n t o account by E i n s t e i n , t h e v i s c o s i t y i n c r e a s e s more r a p i d l y t h a n p r e d i c t e d from t h e r e l a t i o n (1). Considering i n t e r a c t i o n s between two s p h e r e s , B a t c h e l o r /9/ h a s o b t a i n e d a r e l a t i o n up t o t h e second o r d e r i n 9 :

where K is a numerical c o e f f i c i e n t which depends on t h e t y p e o f flow and on t h e importance o f brownian motion ( lowfhigh P e c l e t ' s number r e g i m e s ) . Such a r e l a t i o n a p p l i e s f o r volume f r a c t i o n up t o l o - 15 % f o r s p h e r e s w i t h hydrodynamic i n t e r a c t i o n s o n l y . For l a r g e r v a l u e s of 9, t h e r e is a wide d i v e r s i t y o f phenomenological, semi- e m p i r i c a l o r e m p i r i c a l r e l a t i o n s /lo/.

For s u s p e n s i o n s o f p a r t i c l e s i n t h e c o l l o i d a l range, t h e v i s c o s i t y is i n c r e a s e d by t h e r e p u l s i o n between t h e o v e r l a p p i n g c l o u d s ( c f . I ) and t h e r e s i s t a n c e o f t h e s e c l o u d s t o deformation. As t h e f o r c e s between p a r t i c l e s are g r e a t l y a f f e c t e d by t h e v a r i a t i o n s i n i o n i c c o n c e n t r a t i o n , one o b s e r v e s v e r y important e f f e c t s on t h e v i s c o s i t y

/ll/.

For l a r g e v a l u e s o f @(45 - 60 %), o r d e r e d s t r u c t u r e s can appear i n d i s p e r s i o n o f s p h e r e s , as observed by Hoffman /12/ o r P a t z o l d /13/. The Hoffman's experiments, performed i n a simple s h e a r , show an o r g a n i z a t i o n o f s p h e r e s i n p a r a l l e l p l a n e s i n which t h e r e is an hexagonal o r d e r i n g of s p h e r e s observed by l i g h t d i f f r a c t i o n . Such a s t r u c t u r e r e s u l t s from t h e c o m p e t i t i o n between r e p u l s i v e s f o r c e s on t h e one hand and

Van d e r Waals i n t e r a c t i o n and s h e a r induced s t r e s s e s on t h e o t h e r hand. Above a c r i t i c a l s h e a r r a t e , t h i s s t r u c t u r e is d i s t a b i l i z e d : t h e d i f f r a c t i o n p a t t e r n c h a r a c t e r i s t i c of hexagonal o r d e r d i s a p p e a r s and t h e v i s c o s i t y i n c r e a s e s by o n e o r two o r d e r s o f magnitude. The P a t z o l d ' s experiments show t h a t t h e e x i s t e n c e o f such a s t r u c t u r e is c l o s e l y r e l a t e d t o t h e simple s h e a r i n g motion. I n an e x t e n s i o n n a l flow t h e r e is no r e g u l a r s t r u c t u r e i n t h e suspension and t h e v i s c o s i t y f o r l a r g e ( - 60 % ) v a l u e s of volume f r a c t i o n is h i g h e r i n t h i s t y p e o f flow t h a n i n simple s h e a r i n g motion.

I n the i n t e r m e d i a t e range o f c o n c e n t r a t i o n (20-50 %), s h e a r induced s t r u c t u r e s appear i n s u s p e n s i o n s o f s p h e r e s /14/, / I S / . One o b s e r v e s t h e e x i s t e n c e o f dynamical c l u s t e r s o f p a r t i c l e s even i n t h e c a s e where the o n l y r a l e v a n t i n t e r a c t i o n between s p h e r e s is t h e hydrodynamic one. These c l u s t e r s are observed as w e l l i n real experiment /16/ as i n computer s i m u l a t i o n s /17/ on two-dimensionnal d i s p e r s i o n / l a / . B y analogy w i t h t h e p e r c o l a t i o n t h e o r y /19/, d e Gennes suggested a r h e o l o q i c a l model o f s u s p e n s i o n s /20/ i n which above a c r i t i c a l v a l u e 9' o f t h e volume f r a c t i o n one h a s a two phase system : an i n f i n i t e c l u s t e r on t h e one hand and f l u i d and f i n i t e c l u s t e r s on t h e o t h e r . The r h e o l o q i c a l p r o p e r t i e s o f such a system a r e n o t , a t t h e p r e s e n t time, f u l l y

J O U R N A L DE PHYSIQUE

understood : s t u d i e s on i n t r a - c l u s t e r s and i n t e r - c l u s t e r s c o n t r i b u t i o n s t o t h e v i s c o s i t y , e l a s t i c and v i s c o e l a s t i c p r o p e r t i e s o f c l u s t e r s a r e now i n p r o g r e s s .

111 - SEDIMENTATION

I t is w e l l kown t h a t an i s o l a t e d sphere o f r a d i u s a settles i n an unbounded f l u i d w i t h t h e s t o k e s ' v e l o c i t y U,

where F is t h e buoyancy-corrected weight of t h e sphere and q t h e v i s c o s i t y o f t h e f l u i d . When one c o n s i d e r s a s u s p e n s i o n , t h e average s e t t l i n g v e l o c i t y o f s p h e r e s is lower t h a n t h a t g i v e n by ( 3 ) , even f o r volume f r a c t i o n as low as 1 %. T h i s phenomenon, due t o i n t e r a c t i o n s between s p h e r e s , c a n b e r e p r e s e n t e d b y a n h i n d e r e d s e t t l i n g f u n c t i o n f ( + ) s o t h a t t h e a v e r a g e s e t t l i n g v e l o c i t y is g i v e n by

For l a r g e s p h e r e s w i t h o n l y hydrodynamic i n t e r a c t i o n s , B a t c h e l o r /21/, computed t h e f f u n c t i o n :

The experiments o f B u s c a l l e t c o l l . /22/ on micron-sized s p h e r e s show t h a t t h e f f u n c t i o n is l i n e a r i n ^$J ( w i t h a c o e f f i c i e n t o f the o r d e r o f 5 r a t h e r t h a n 6 . 5 5 ) up t o 10%.

The t h e o r e t i c a l a n a l y s i s h a s t o t a k e i n t o account t h e m u l t i p a r t i c l e i n t e r a c t i o n and t h e s p a t i a l d i s t r i b u t i o n o f t h e sedimenting p a r t i c l e s which is n o t known a p r i o r i b e i n g i t s e l f a p a r t o f t h e problem. So t h e B a t c h e l o r ' s r e s u l t ( 5 ) assumes t h a t , f o r low v a l u e s o f 4, t h e f i r s t c o r r e c t i o n t o Stokes's law r e s u l t s f o r p a i r w i s e i n t e r a c t i o n s i n a random d i s t r i b u t i o n . Non hydrodynamic i n t e r a c t i o n s between p a r t i c l e s as Van d e r Waals a t t r a c t i v e f o r c e s c a n c a u s e a non-random d i s t r i b u t i o n w i t h a n e x c e s s o f c l o s e p a i r s o f s p h e r e s ^/23/. As f o r t h e v i s c o s i t y , one o f t h e q u e s t i o n t o be s o l v e d , is t h e e x i s t e n c e o f a m i c r o s c a l e s t r u c t u r e i n a sedimenting s u s p e n s i o n due t o m u l t i p a r t i c l e hydrodynamic and/or non hydrodynamic i n t e r a c t i o n s .

Assuming t h a t f ( $ J ) depends o n l y o n t h e volume f r a c t i o n and monotonically d e c r e a s e s w i t h @, Kynch /24/ used kinematic-wave t h e o r y t o d e s c r i b e t h e s h i f t i n g o f +-

d i s c o n t i n u i t i e s between t h e s u s p e n s i o n s and t h e c l e a r f l u i d at t h e t o p on t h e one hand and between s u s p e n s i o n s and t h e sediment l a y e r a t t h e bottom, on t h e o t h e r hand. The v e l o c i t y o f t h e upper d i s c o n t i n u i t y is used i n measurements o f t h e average s e t t l i n g v e l o c i t y /22/, /25/ from which it is p o s s i b l e t o propose e m p i r i c a l e x p r e s s i o n s o f t h e f f u n c t i o n . The above-mentionned p o i n t s t o g e t h e r w i t h sediment at i o n i n i n c l i n e d v e s s e l , p o l y d i s p e r s e s u s p e n s i o n s , and l a t e r a l s e g r e g a t i o n a r e t h e s u b j e c t s o f an e x c e l l e n t review by Davis and Acrivos /26/.

IV- AGGREGATION PROCESSES

As mentionnee i n I , a g g r e g a t i o n p r o c e s s c a n o c c u r i n a s u s p e n s i o n . When t h e system i s m a c r o s c o p i c a l l y at r e s t t h e brownian motion may b r i n g t o g e t h e r two p a r t i c l e s which may s t i c k under t h e a c t i o n o f a t t r a c t i v e i n t e r a c t i o n . The rate o f formation o f p a i r s o f p a r t i c l e s and more t h e growth o f c l u s t e r s o f p a r t i c l e s h a s been s t u d i e d f i r s t by Smoluchowski / 2 7 / . Recent computer s i m u l a t i o n s , i n i t i a t e d by t h e work o f Witten and s a n d e r /28/, showed t h a t , i n t h i s d i f f u s i o n - l i m i t e d a g g r e g a t i o n ( D.L.A. ), the c l u s t e r s a r e n o t v e r y compact. T h e i r d e n d r i t i c shapes p r e s e n t s e l f - s i m i l a r p r o p e r t i e s which can be expressed by a f r a c t a l dimension dF r e l a t e d t o t h e E u c l i d i a n dimension d ( t h e dimension o f t h e s p a c e i n which the c l u s t e r s growth) by t h e approximate r e l a t i o n /29/.

dimensions) viewpoint as from tempors.1 e v o l u t i o n (time-dependant c l u s t e r - s i z e d i s t r i b u t i o n ) . Real experiments on a g g r e g a t i o n i n a system at r e s t have been r e c e n t l y performed on a g g r e g a t i n g p r o t e i n s /32/, g o l d c o l l o i d s /33/ o r macroscopic s p h e r e s /34/.

When t h e d i s p e r s i o n is submitted t o a s h e a r , t h e convection motion of the suspending f l u i d s p l a y s an important role i n t h e a g g r e g a t i o n p r o c e s s e s . Two l i m i t i n g regimes appear : t h e p r e v i o u s one ( d i f f u s i o n - l i m i t e d ) where t h e a g g r e g a t i o n is s o l e l y due t o brownian motion ( p e r i k i n e t i c a g g r e g a t i o n ) and, at t h e o p p o s i t e , t h e o r t h o k i n e t i c a g g r e g a t i o n s o l e l y due t o s h e a r . I n t h i s last c a s e t h e hydrodynamic f o r c e s and t o r q u e s a c t i o n on t h e s p h e r e s c o n t r o l , t o g e t h e r with t h e o t h e r i n t e r a c t i o n s (Van d e r Waals, d o u b l e - l a y e r ) , the growth o f a g g r e g a t e s . The e f f e c t o f t h e s h e a r i s double. On t h e one hand, t h e c o l l i s i o n frequency and hence t h e p o s s i b l e b u i l d i n g o f a d o u b l e t v a r i e s l i n e a r l y w i t h the s h e a r r a t e G, b u t on the o t h e r hand t h e mean l i f e - t i m e o f a d o u b l e t o f close s p h e r e s f o r which a t t r a c t i v e f o r c e s c a n p l a y a r o l e v a r i e s l i k e G-'. Then t h e rate o f formation o f p a i r s may be expressed /20-b/ by :

where a is a s t i c k i n g p r o b a b i l i t y r e l a t e d t o t h e s h e a r r a t e and t o a c h a r a c t e r i s t i c t i m e

T o f a t t r a c t i v e f o r c e s by

Values 0 . 5 and 0.18 of t h e m exponent have been r e s p e c t i v e l y o b t a i n e d by d e Gennes

/20--b/ and Van d e Ven and Mason /35/. From ( 7 ) and ( a ) , one can s e e t h a t t h e r a t e o f a s s o c i a t i o n is a n i n c r e a s i n g f u n c t i o n o f G .

V- CONCLUSION

I n t h i s paper, we have p u t forward t h e main p h y s i c a l q u a n t i t i e s r e l e v a n t t o t h e p r o p e r t i e s o f c o n c e n t r a t e d s u s p e n s i o n s t h e importance o f which is c o n s i d e r a b l e i n n a t u r a l and i n d u s t r i a l p r o c e s s e s . When t h e volume f r a c t i o n i n t h e suspension is l a r g e , t h e i n t e r a c t i o n s between p a r t i c l e s p l a y a n important r o l e i n t h e r h e o l o g i c a l p r o p e r t i e s o f t h e system. Depending on t h e t y p e o f flow, r e g u l a r o r random s t r u c t u r e s c a n appear i n t h e suspension. Such s t r u c t u r e s c l e a r l y appear i n a g g r e g a t i o n p r o c e s s e s and are s u s p e c t e d i n sedimenting d i s p e r s i o n . The i n f l u e n c e o f t h e s e s t r u c t u r e s on t h e r h e o l o g i c a l p r o p e r t i e s is u n d o u b t l e s s important b u t , up t o now, l a r g e l y misunderstood.

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