STRUCTURE OF FRACTAL COLLOIDAL AGGREGATES FROM SMALL ANGLE X-RAY SCATTERING

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STRUCTURE OF FRACTAL COLLOIDAL AGGREGATES FROM SMALL ANGLE X-RAY

SCATTERING

D. Schaefer, J. Martin, K. Keefer

To cite this version:

D. Schaefer, J. Martin, K. Keefer. STRUCTURE OF FRACTAL COLLOIDAL AGGREGATES

FROM SMALL ANGLE X-RAY SCATTERING. Journal de Physique Colloques, 1985, 46 (C3),

pp.C3-127-C3-135. �10.1051/jphyscol:1985311�. �jpa-00224628�

STRUCTURE O F FRACTAL C O L L O I D A L AGGREGATES FROM SMALL A N G L E X-RAY SCATTERI NGr

D.W. Schaefer, J.E. Martin and K.D. Keefer

Sandia National ïkboratofies, Albuquerque, New Mexico 87185, U.S.A.

Résumé - La s t r u c t u r e d e s a g r é g a t s de s i l i c e c o l l o i d a l e e s t é t u d i é e p a r d i f f u s i o n d e s r a y o n s X aux p e t i t s a n g l e s . I l s ap- p a r a i s s e n t r a m i f i é s e t p e u v e n t ê t r e c o n s i d é r é s comme d e s o b j e t s f r a c t a l s d e d i m e n s i o n n a l i t é D . D e s t obtenu à p a r t i r de l a p e n t e du f a c t e u r d e s t r u c t u r e e n f o n c t i o n du v e c t e u r de d i f f u s i o n K d a n s l e régime KR >>1 >> Ka. Rg e s t l e r a y o n d e g y r a t i o n d e l ' a g r é g a t t a n d i s que a g e s t l e rayon du "monomère". Nous o b t e n o n s D

2 , 0 + 0 , 1 5 . Pour Ka

1, nous o b s e r v o n s un changement de comporte- ment e t D d e v i e n t é g a l à 3 e n a c c o r d a v e c l e f a i t que l e mono- mère e s t d e n s e .

A b s t r a c t - The s t r u c t u r e of a g g r e g a t e s o f c o l l o i d a l s i l i c a i s s t u d i e d by s m a l l a n g l e x - r a y s c a t t e r i n g . These a g g r e g a t e s have a r a m i f i e d a p p e a r a n c e . The a g g r e g a t e s a r e a p p r o p r i a t e l y d e s c r i b - ed by f r a c t a l geometry a n d , t h e r e f o r e , by a f r a c t a l d i m e n s i o n , D . D i s d e t e r m i n e d from t h e s l o p e of t h e s t a t i c s t r u c t u r e f a c t o r i n t h e regime K R 4 >> 1 >> Ka where Rg i s an a v e r a g e r a d i u s of g y r a t i o n of t h e a g g r e g a t e s , 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

"monomer" and K i s t h e magnitude of t h e s c a t t e r i n g v e c t o r .

We f i n d D

2.0 + - 1 5 . A t Ka

1, a c r o s s o v e r t o D = 3 i s found, c o n s i s t e n t w i t h t h e d e n s e s t r u c t u r e o f t h e c o l l o i d a l monomer.

1. INTRODUCTION

I n t h e o r d e r l y world of c o l l o i d a l c r y s t a l s random c o l l o i d a l a g g r e g a t e s a r e c o n s i d e r e d a n u i s a n c e t o t h e e x p e r i m e n t a l i s t . These o b j e c t s , how- e v e r , a r e of c o n s i d e r a b l e i n t e r e s t t o t h o s e who wish t o u n d e r s t a n d t h e s t r u c t u r e o f random s y s t e m s i n t e r m s o f e q u i l i b r i u m s t a t i s t i c a l - m e c h - a n i c a l models o r k i n e t i c growth p r o c e s s e s . I n t h i s p a p e r we s t u d y t h e s t r u c t u r e of aqueous s i l i c a a g g r e g a t e s by s m a l l a n g l e x - r a y s c a t t e r i n g

( S A X S ) . I n p a r t i c u l a r , t h e f r a c t a l d i m e n s i o n , D , o f t h e c l u s t e r s i s d e t e r m i n e d from t h e s l o e of t h e s c a t t e r e d x - r a y i n t e n s i t y , I ( K ) , i n t h e power-law r e g i m e . l S P The r e s u l t s a r e i n c o n s i s t e n t w i t h c u r r e n t k i n e t i c models, b u t may i n s t e a d r e f l e c t e q u i l i b r i u m s t a t i s t i c s .

C o l l o i d a l a g g r e g a t e s a r e formed u n d e r t h e o p p o s i t e c o n d i t i o n s r e q u i r e d t o produce c o l l o i d a l c r y s t a l s . I n charged s y s t e m s , a g g r e g a t e s form when t h e r e p u l s i v e coulomb p o t e n t i a l between c o l l o i d a l p a r t i c l e s i s r e - duced t o t h e p o i n t t h a t t h e r e i s a n a p p r e c i a b l e p r o b a b i l i t y t h a t t h e i n t e r p a r t i c l e p o t e n t i a l b a r r i e r i s c r o s s e d . C l e a r l y , a g g r e g a t i o n i s f a v o r e d when t h e pH i s n e a r t h e i s o e l e c t r i c p o i n t , r e d u c i n g t h e s u r f a c e c h a r g e , and / o r when t h e i o n i c s t r e n g t h i s i n c r e a s e d , t h e r e b y r e d u c i n g t h e Debye s c r e e n i n g l e n g t h . F i g u r e 1 shows a t y p i c a l s i l i c a a g g r e - g a t e .

* ~ h i s work was performed at Sandia National Laboratories by the U.S. Department of Energy under Contract Number DE-AC04-76-DP00789.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985311

C3-128 JOURNAL DE PHYSIQUE

F i g . 1 E l e c t r o n m i c r o g r a p h o f a c o l l o i d a l s i l i c a c l u s t e r . The b a r i s Z S O O A .

Two d i s t i n c t c l a s s e s of a g g r e g a t e s a r e a n t i c i p a t e d . The f i r s t t y p e , d e n s e homogeneous c l u s t e r s , might o c c u r when b o t h t h e r e p u l s i v e b a r r i e r and t h e a t t r a c t i v e w e l l a r e comparable t o kT ( s e e F i g . 2 ) . Under t h e s e c o n d i t i o n s , a n i r r e g u l a r c l u s t e r c a n a n n e a l t o a dense s t r u c t u r e

( a p r o c e s s s i m i l a r t o Ostwald r i p e n i n g ) . I f , on t h e o t h e r hand, t h e

a t t r a c t i v e w e l l i s d e e p , t h e n t h e system may n o t r e a c h e q u i l i b r i u m

s i n c e once amonomer i s s t u c k , i t w i l l n e v e r g e t l o o s e a g a i n . Under

t h e s e c o n d i t i o n s , we e x p e c t h i g h l y r a m i f i e d o b j e c t s formed by 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 , a p r o c e s s f i r s t s t u d i e d by F o r r e s t and ~ i t t e n ?

These o b j e c t s a r e c a l l e 6 f r a c t a l c l u s t e r s o r s i m p l y f r a ~ t a l s . ~

F i g . 2 Schematic r e p r e s e n t a t i o n o f t h e p o t e n t i a l between charged c o l l o i d a l p a r t i c l e s . The u p p e r c u r v e r e p r e - s e n t s a s t a b l e system whereas t h e lower c u r v e l e a d s t o r a p i d a g g r e g a t i o n .

Because of t h e s t r o n g a n a l o g y between branched polymers and c o l l o i d a l a g g r e g a t e s , i t i s p o s s i b l e t o u s e t h e language of polymer p h y s i c s t o d e s c r i b e them. I n t h e c o l l o i d a l s y s t e m , 'monomers' a r e t h e i n d i v i d u a l s p h e r i c a l p a r t i c l e s ; t h e c l u s t e r i t s e l f i s a polymer. The s t r u c t u r e of t h i s c l u s t e r c a n be a n a l y z e d i n terms o f b r a n c h i n g , a s i n c o n v e n t i o n - a l polymers.

The c o l l o i d a l a g g r e a t e i s an u n u s u a l polymer w i t h an enormous monomer dimension ( a - 1 0 0 f ) . J u s t a s t h e huge l a t t i c e c o n s t a n t i n c o l l o i d a l c r y s t a l s opened a new r e a l m o f " s o l i d - s t a t e " p h y s i c s , we a n t i c i p a t e a new domain of "polyner" p h y s i c s c o n c e i v e d by a u n i o n o f c o l l o i d a l and polymer c o n c e p t s . A s i n t h e c a s e of c o l l o i d a l c r y s t a l s , t h e s e new polymers a r e c o n v e n i e n t l y s t u d i e d by s t a t i c and dynamic l i g h t s c a t t e r - i n g .

2 . THE EXPERIMENT

The s i l i c a system was chosen f o r s e v e r a l r e a s o n s . F i r s t , i t i s n e c e s - s a r y t o u s e r a t h e r s m a l l monomers ( r a d i u s

a ) t o r e a c h t e domain K << 1 p a r t i c u l a r l y f o r SAXS e x p e r i m e n t s where KMIN = O . 1 I f l a r g e p a r t i c l e s ( e . g . , p o l y s t y r e n e s p h e r e s ) a r e u s e d , t h e . x - r a y domain

i s c o m p l e t e l y dominated by t h e form f a c t o r of t h e monomer and y i e l d s no i n f o r m a t i o n on t h e geometry o f t h e c l u s t e r . The s i l i c a system i s a l s o d e s i r a b l e because i t i s commercially a v a i l a b l e , w e l l c h a r a c t e r - i z e d and o f f e r s good x - r a y c o n t r a s t .

A g g r e g a t i o n i s i n i t i a t e d by changing b o t h t h e pH and he i o n i c s t r e n g t h .

B

For t h e SAXS work, ~udoxTM SM (nominal d i a m e t e r = 70 ) was d i l u t e d t o 1% volume < .SM NaCl. The pH was t h e n reduced t o 5 . 5 w i t h

.SM HC1. The d i l u t i o n was performed s u c h t h a t t h e f i n a l NaCl c o n c e n t -

r a t i o n was .SM. Under t h e s e c o n d i t i o n s , a g g r e g a t i o n t o o k p l a c e o v e r

a two day p e r i o d . I n t h e l a t t e r s t a g e s of a g g r e g a t i o n , t h e l a r g e

c l u s t e r s sedimented u n d e r g r a v i t y . The SAXS e x p e r i m e n t s were perform-

e d p r i o r t o v i s i b l e s e d i m e n t a t i o n .

C3-130 JOURNAL DE PHYSIQUE

A f t e r a b o u t 20 h o u r s t h e c l u s t e r s became t o o l a r g e t o r e s o l v e w i t h -

o u r SAXS a p p a r a t u s . I n o r d e r t o o b t a i n some measure of t h e c l u s t e r s i z e a t t h a t t i m e , dynamic l i g h t s c a t t e r i n g measurements were perform- ed on h i g h l y d i l u t e d samples ( - . 0 1 % ) . S i n c e i n t e r p r e t a t i o n of l i g h t s c a t t e r i n g d a t a i s complex f o r t h e s e v e r y l a r g e , p o l y d i s p e r s e c l u s t - e r s , we measured a n e f f e c t i v e d i f f u s i o n c o n s t a n t D e f f = ~ K - Z , which confirmed t h e e x i s t e n c e of l a r g e s t r u c t u r e s . Here r i s t h e mean decay r a t e measured from t h e i n i t i a l s l o p e of t h e s c a t t e r e d i n t 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 .

SAXS e x p e r i m e n t s were performed on a Kratky SAXS system w i t h a r o t a t - i n g anode s o u r c e and a p o s i t i o n - s e n s i t i v e d e t e c t o r . One o f u s (KDK) a d a p t e d a Anton P a a r Kratky compact camera t o a Rigaku mode1 RU200 1 2 KW x - r a y g e n e r a t o r . A TEC c a r b o n f i b e r s c i n t i l l a t i o n d e t e c t o r i s 25 cm from t h e sample. O p e r a t i n g a t 7 KW, t h e c u r v e s shown i n Fig.3 were o b t a i n e d i n 500 -1000 s e c . Because t h e a p p a r a t u s i s s t i l l u n d e r development, t h e d a t a were n o t c o r r e c t e d f o r e i t h e r d e t e c t o r l i n e a r i t y o r s e n s i t i v i t y . A t t h e p r e s e n t s t a g e o f development, s u c h c o r r e c t i o n s a r e s m a l l e r t h a n o u r a b i l i t y t o measure them: a s o l v e n t background, however, was s u b t r a c t e d . Both t h e a g g r e g a t i o n and t h e SAXS e x p e r i - ments were psrformed a t room t e m p e r a t u r e s , 2 4 + 3' C .

S i n c e t h e s e e x p e r i m e n t s were performed w i t h t h e l i n e - s o u r c e geometry c h a r a c t e r i s t i c of a Kratky camerab , t h e s i m p l e p r o d u c t of t h e form f a c t o r and t h e s t r u c t u r e f a c t o r

ISM(K) f P(K1 S(K) (2)

was n o t d i r e c t l y measured. Here 1 (K) i s t h e s l i t - s m e a r e d measured i n t e n s i t y , P(K) i s t h e form f a c t o r S M w h i c h r e f l e c t s t h e geometry of a s i n g l e monomer p a r t i c l e , and S(K) i s t h e s t r u c t u r e f a c t o r , which de- pends on i n t r a c l u s t e r c o r r e l a t i o n s . R a t h e r , a s l i t - s m e a r e d v e r s i o n of S(K) P(K) was measured b e c a u s e , a t any p o i n t on t h e d e t e c t o r , p h o t - ons a r e c o l l e c t e d which a r e s c a t t e r e d t h r o u g h a d i s t r i b u t i o n of s c a t - t e r i n g a n g l e s . I t t u r n s o u t t h a t i n t h e l i m i t KR

O: t h e measured p r o f i l e i s u n d i s t o r t e d and R can be a c c u r a t e l y d g t e r m i n e d from t h e i n i t i a l decay of ISM(K). I n r e g i m e s where power-law decay i s ex e c t - g

- t

ed a s i m p l e r e l a t i o n a l s o e x i s t s f o r t h e s l i t - s m e a r e d i n t e n s i t y .

ISM(K) K P(K) S(K1 ( 3 )

I t i s w e l l known t h a t 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) shows power law d e c a y 7 , 8 f o r s e l f - s i m i l a r f r a c t a l o b j e c t s

s(K)-., K - ~ ; R - ~ << K << a-' (4)

ISI4 (KI

K 1 - D _{( 5 )}

Note t h a t i n t h e l i m i t KR << 1, P ( K )

1. I n t h e o p p o s i t e l i m i t ,

>> 1, SfK) i s a c o n s t a f i t and PfK) d e c a y s a s K - ~ c h a r a c t e r i s t i c of y g c o l l o i d a l p a r t i c l e w i t h s h a r p b o u n d a r i e s . l i 6 I n t h i s regime we

e x p e c t - 3

lSM(K)mK

3 . RESULTS

F i g u r e 3 shows t h e development of t h e s c a t t e r i n g c u r v e s d u r i n g t h e

c o u r s e of a g g r e g a t i o n i n 1% s i l i c a . The c u r v e s a r e m u l t i p l i e d by an

a r b i t r a r y c o n s t a n t t o s p r e a d them o u t on t h e g r a p h . A t p r e s e n t , we

a r e u n a b l e t o make d i r e c t comparison between t h e i n t e n s i t i e s ( i . e . ,

I S M ( 0 ) ) b e c a u s e of l o n g term f l u c t u a t i o n s i n b o t h s o u r c e and d e t e c t o r .

The c u r v e s show t h e q u a l i t a t i v e f e a t u r e s e x p e c t e d f o r an a g g r e g a t i n g

system. The l o w e s t c u r v e , measured 1 hour a f t e r t h e pH was a d j u s t e d

t h e l o g - l o g p l o t ) a r a d i u s o f g y r a t i o n o f S y ! ~ was o b t a i n e d , t h i s c o r - r e s p o n d i n g t o a h a r d s p h e r e d i a m e t e r o f 142A. T h i s v a l u e i s t w i c e t h e n o m i n a l p a r t i c l e d i a m e t e r o f Ludox SM, i n d i c a t i n g some a g g r e g a t i o n i n t h e f e e d s t o c k . P r e s u m a b l y t h e n , t h e a c t u a l 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 from e x i s t i n g s m a l l c l u s t e r s o f d i m e r i c d i m e n s i o n s .

F i g . 3 . S l i t - s m e a r e d SAXS i n t e n s i t y p r o f i l e s f o r s i l i c a c l u s t e r s d u r i n g a g g r e g a t i o n . t i s t h e t i m e i n c r e m e n t from i n i t i - a t i o n o f t h e a g g r e g a t i o n . Curves a r e d i s p l a c e d f o r c l a r i t y . The b r e a k i n t h e c u r v e s o c c u r s a t t h e n o m i n a l r a d i u s o f g y r a t i o n o f t h e monomer.

A s a g g r e g a t i o n p r o c e e d s , t h e i n i t i a l c u r v a t u r e o f ISM(K) i n c r e s e s , c o n - s i s t e n t wigh c l u s t e r g r o w t h . R was f o u n d t o i n c r e a s e from 5 5 1 a t 1 h o u r t o 66A a t 4 h o u r s and 9 2 a 0 8 t 1 9 . 5 h o u r s . S i n c e Our e q u i p m e n t i s r e s o l u t i o n l i m i t e d a t R = l O O A we w e r e u n a b l e t o f o l l o w c l u s t e r g r o w t h beyond 20 h o u r s by SAXS! T h i s l a c k o f r e s o l u t i o n i s a p p a r e n t from t h e a b s e n c e o f a l i m i t i n g v a l u e f o r ISM(K+ O) i n t h e u p p e r c u r v e s .

I n t h e l a t e r s t a g e s o f g r o w t h , t h e dynamic s t r u c t u r e f a c t o r , S ( K , t ) , was m e a s u r e d by l i g h t s c a t t e r i n g f r o m a d i l u t e d s a m p l e . F i g u r e 4

O

1

shows t h e m e a s u r e d i n t 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 a t K

1 . 8 8 x ~ O - ~ P . : The c o r r e l a t i o n f u n c t i o n i s n o n e x p o n e n t i a l , r e f l e c t i n g a b r o a d d i s t r i - b u t i o n o f c l u s t e r s i z e . From t h e i n i t i a l s l o p e o f t h e c o r r e l a t i n f u n c t i o n a n e f f e c t i v e Z-averaged hydrodynamic r a d i u s o f 2 n 1 0 3 1 ^was

o b t a i n e d . The a c t u a l 2 - a v e r a g e d hydrodynamic r a d i u s i s c o n s i d e r a b l y

l a r g e r t h a n t h i s v a l u e s i n c e m e a s u r e m e n t s w e r e n o t made i n t h e l i m i t

KR

O . The p o i n t i s t h a t t h e c l u s t e r s w e r e v e r y l a r g e when t h e

t o 5 c u r v e s o f F i g . 3 w e r e t a k e n .

JOURNAL DE PHYSIQUE

0.100

O

8

24 32 40 48 56 64 CHANNEL NUMBER

F i g . 4 . T y p i c a l i n t 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 measured a f t e r A t = 30 h r s . The d e l a y p e r c h a n n e l i s 8 x 10-5s. The d a t a a r e n o r m a l i z e d and a random c o i n c i d e n c e background

i s s u b t r a c t e d .

The t h r e e e a r l y c u r v e s show a d i s t a n t b r e a k a t a - 1, where a i s t h e

f

e f f e c t i v e r a d i u s of g y r a t i o n o f t h e monomer (55 ).The f a c t t h a t t h i s b r e a k p o i n t i s i n d e p e n d e n t of time i n d i c a t e s t h a t t h e p a r t i c l e s r e - main d i s t i n c t on a 50A s c a l e . I f t h e a g g r e g a t e s were d e n s e o r i f r i p -

e n i n g o c c u r r e d , o r even i f d e n s e c l u s t e r s of more t h a n monomeric p r o - p o r t i o n s e x i s t e d , t h i s b r e a k would move t o s m a l l e r K.

For Ka >> 1, a l 1 t h e c u r v e s d i s p l a y power-law b e h a v i o r w i t h a n exponent of -3.0

. I O . T h i s exponent i s c o n s i s t e n t w i t h P o r o d ' s law6 which p r e d i c t s a s l o p e of -4 P(K) f o r any s t r u c t u r e w i t h s h a r p b o u n d a r i e s on a s c a l e K-1. The o b s e r v e d s l o p e o f -3 i s e x p e c t e d when P o r o d ' s law i s o b s e r v e d i n s l i t geometry [eq. (6)] .

The SAXS d a t a a l s o show power-law b e h a v i o r f o r Ka << 1, a f t e r t h e c l u s t - e r s have grown w e l l beyond t h e r e s o l u t i o n l i m i t o f t h e i n s t r u m e n t . Pow- e r - l a w s t r u c t u r e f a c t o r s a r e c o n s i s t e n t w i t h r a m i f i e d c l u s t e r s o r f r a c t - a l o b j e c t s . From t h e s e d a t a , t h e f r a c t a l dimension, D, of t h e c l u s t e r s was d e t e r m i n e d t o be D

2.0 + .15. L i g h t s c a t t e r i n g measurements p e r - formed i n c o l l a b o r a t i o n w i t h t h e S a n t a B a r b a r a group7 gave D

2.11 + . 0 3 , c o n s i s t e n t w i t h t h e s e x - r a y r e s u l t s .

4. DISCUSSION

To o u r knowledge, o n l y two models e x i s t which p r e d i c t t h e f r a c t a l s t r u c t u r e o f c o l l o i d a l a g g r e g a t e s : ~ i t t e n - ~ a n d e r 8 d i f f u s i o n l i m i t e d ag- g r e g a t i o n (DLA) and c l u s t e r a g r e g a t i o n (CA) s t u d i e d by b o t h ~ e a k i n ~ and Kolb, B o t e t and J u l l i e n . l a Both mode1 a r e k i n e t i c i n t h e s e n s e t h a t t h e s t r u c t u r e s a r e n o t a t thermodynamic e q u i l i b r i u m .

I n DLA, c l u s t e r s grow p u r e l y from monomers, which approach t h e c l u s t e r

w i t h a random-walk t r a j e c t o r y . 8 , 9 The monomer i s presumed t o s t i c k i f

e x t r e m i t i e s o f t h e c l u s t e r , and t h u s , open, r a m i f i e d g e o m e t r i e s ( a s i n Fig.1) d e v e l o p . The f r a c t a l dimension f o r DLA i s 2.5

. O 2 i n t h r e e d i m e n s i o n s . We o b s e r v e a c o n s i d e r a b l y s m a l l e r v a l u e t h a n 2.5 s o t h i s mode1 needs m o d i f i c a t i o n .

C l u s t e r a g g r e g a t i o n i s a v a r i a t i o n of DLA where c l u s t e r s a p e r m i t t e d t o grow from e x i s t i n g c l u s t e r s a s w e l l a s from monomers.

I n CA, D i s reduced s u b s t a n t i a l l y b e c a u s e two f r a c t a l s a r e e x t r e m e l y u n l i k e l y t o p e n e t r a t e w i t h o u t c o n t a c t . CA g i v e s 1 5 ~ 1.78 i n d

3. Although t h i s v a l u e i s i n c o n s i s t e n t w i t h o u r o b s e r v a t i o n s , o t h e r s have o b s e r v e d D E 1.6-1.8 f o r smoke p a r t i c l e s 3 and g o l d c o l l o i d s . l l I n t h e s e e x p e r i - m e n t s , D was measured from e l e c t r o n microscope images of c o l l a p s e d c l u s t - e r s and i t i s p o s s i b l e t h a t p r e p a r a t i o n p r o c e d u r e s a l t e r e d t h e o b s e r v e d D. Of c o u r s e , i t i s a l s o p o s s i b l e t h a t d i f f e r e n t growth models a p p l y .

I n our system we b e l i e v e t h a t c l u s t e r growth i s n o t a k i n e t i c p r o c e s s . I n s t e a d , we u s e an a n a l o g of F l o r y t h e o r y 1 2 f o r polymers t o d e v e l o p a m e a n - f i e l d , e q u i l i b r i u m p r e d i c t i o n f o r D .

~ l o r y l 2 p r e d i c t e d t h e s t r u c t u r e of s w o l l e n l i n e a r polymers from a mean- f i e l d a n a l y s i s of t h e s e l f - a v o i d i n g walk problem. H i s method was t o b a l a n c e t h e e n t r o p i c ( e 1 a s t i c ) c o n t r i b u t i o n s t o t h e f r e e e n e r g y , F, w i t h

t h e e n t h a l p i c c o n t r i b u t i o n due t o r e p u l s i v e f o r c e s between monomers.

The e n t r o p i c c o n t r i b u t i o n f a v o r s c o n f i g u r a t i o n s w i t h i d e a l o r random- walk c o n f i g u r a t i o n s , whereas t h e e n t h a l p i c term f a v o r s s w e l l i n g . With

t h e e n t r o p i c term c a l c u l a t e d f o r Gaussian s t a t i s t i c s and t h e excluded volume c o n t r i b u t i o n c a l c u l a t e d i n t h e m e a n - f i e l d a p p r o x i m a t i o n , F l o r y

f i n d s D = 5/3 f o r l i n e a r c h a i n s . T h i s v a l u e h a s been o b s e r v e d i n numer-

D U S

systems.12

F l o r y t h e o r y h a s a l s o been a p p l i e d t o s e l f - a v o i d i n randomly branched c h a i n s ( l a t t i c e a n i m a l s ) by I s a a c s o n and Lubensky.f3 They f i n d D= 2 . O f o r d

3 and once a g a i n t h i s v a l u e i s c o n s i s t e n t w i t h t h e o n l y measure- ment of D f o r branched polymers .2

F l o r y t h e o r y haç c e l e b r a t e d s h o r t c o m i n g s .16 I n p a r t i c u l a r , t h e e n t r o p - i c c o n t r i b u t i o n t o F i s o v e r e s t i m a t e d s i n c e t h e Gaussian a p p r o x i m a t i o n i s c l e a r l y i n c o r r e c t f o r s w o l l e n c h a i n s . T h i s e r r o r , however, i s com- p e n s a t e d by t h e n e g l e c t of f l u c t u a t i o n s i n t h e m e a n - f i e l d approxima- t i o n f o r t h e e x c l u d e d volume term. I n t h e c a s e of branched c h a i n s ? a f u r t h e r d i f f i c u l t y o c c u r s i n t h a t t h e e l a s t i c term i s based on t h e Gaussian a p p r o x i m a t i o n f o r a randomly branched o b j e c t on a Cayley t r e e . Such a s t r u c t u r e i s c o m p l e t e l y n o n i n t e r s e c t i n g ( c o n t a i n s no c i r c u i t s ) and, n e g l e c t i n g t h e e n t h a l p i c term, h a s t h e u n p h y s i c a l c h a r a c t e r i s t i c D

4 . I n s p i t e of t h e s e s h o r t c o m i n g s , F l o r y t h e o r y h a s been h i g h l y s u c c e s s f u l and, u n t i l r e c e n t l y , a t t e m p t s t o improve on t h e a s s u m p t i o n s have degraded agreement w i t h e x p e r i m e n t . We a p p l y t h e F l o r y argument t o c o l l o i d s , and o b t a i n t h e l a t t i c e animal e q u a t i o n s of I s a a c s o n and

~ u b e n s k ~ l ~ .

For t h e c o l l o i d problem, t h e e n t h a l p i c c o n t r i b u t i o n h a s t h e same o r i g - i n a s i n t h e polymer problem. Namely, t h e r e p u l s i v e f o r c e s between monomers l e a d t o a c o n t r i b u t i o n t o F which i s p r o p o r t i o n a l t o t h e mean

number of b i n a r y c o n t a c t s w i t h i n t h e p a r t i c l e 2 3

Fex . v o l . /kT

V N / R ( 7 )

where N i s t h e d e g r e e o f p o l y m e r i z a t i o n and V i s t h e s o - c a l l e d e x c l u d - ed volume. For h a r d s p h e r e s of r a d i u s r , V

8xr3.

For t h e e n t r o p i c c o n t r i b u t i o n t o F we assume a form i d e n t i c a l t o

F l o r y ' s e l a s t i c term b u t t h i s term i s n o t e l a s t i c i n o r i g i n . Presum-

C3-134 JOURNAL DE PHYSIQUE

ably the colloidal aggregate is rigid. Rather, this term is the log of the probability distribution, P(R) characterizing branching in a

completely non-interacting system. Following Isaacson and ~ u b e n s k ~ , ~ ~ let P(R) be the distribution for branched structures in a Cayley tree

(where R ~ 1 1 4 ) . ^Then,

Combining (7) and (a), \-. ,'

VN

F/kT

+ - + ...

R ⁽⁹⁾

1

Minimization of (9) leads to,

Equation (10) is one way to define the fractal dimension. Since the value D

2 agrees with our x-ray scattering results, Our data are consistent with the lattice animal interpretation.

The critical assumption of this calculation is the form of eq.(8).

This form results from the excluded volume effect. That is, once a branched structure has formed, we expect an equilibrium distribution of monomers which decreases toward the center of the cluster. If there is a residual potential barrier between monomers (curve b of Fig.2), then monomers will be excluded from the denser areas of the cluster.

Since few monomers are present near the core, growth will occur on the perimeter even at equilibrium.

Daoud and ~ o a n n ~ l 4 have considered the case V

O , where the third order term must be included in (9). In this situation (corresponding

to theta solutions for polymers) they find D

16/7

2.28. deGennes 17 identifies the V

O case with percolation at threshold, so it is like- ly that Daoud and Joanny's result approximates the structure of span- ning percolation clusters for which D

2.5. At any rate, we expect

D > 2 for lattice animals as the excluded volume is reduced.

When the above analysis is correct, colloidal growth leads to equili- brium structures. In our system, the s-ticking probability may be low enough to allow penetration of monomers into the crevices of the ag- gregates. In aduition, the repulsive coulomb forces guarantee the required self-avoiding rule for growth. Thus, both the penetration factor and the growth rule depend on the existence of repulsive forces between monomers. These same forces would also eliminate contacts between clusters so growth always occurs by monomers sticking on clust-

ers. These conclusions imply that the interparticle potential must have the form shown in curve (b) of Figure 2. Curve (a) precludes ag- gregation on a reasonable time scale. For (c), on the other hand, either DLA or dense clusters result depending on the depth of the well.

In the model presented here, both short-range attraction and long-range

repulsion are nece,jsary. Without short range attraction, aggregation

would not occur. On the other hand, if long-range repulsion were ab-

sent, the self-avoiding structure would collapse to a dense particle

with D

3. It should be clear that the details of the potential are

very important to a proper analysis of this problem. In view of the

acknowledged limitations of Flory theory, however, such problems may

be secondary within the model discussed here.

Chem.Soc. ,Div.Chem. 239 (1983)

2. D. W. Schaefer and J. G. Curro, Ferroelectrics, g , 49 (1980) 3. S. R. Forrest and T. A. Witten, J. Phys.A., s, LI09 (1979) 4. B. Mandelbrot, Fractals, Form, Chance and Dimension (Freeman,

San Francisco, 1977)

5. J. E. Martin and K. D. Keefer, to be published 6. G. Porod, Kolloid 2. ,124 83 (1951)

7. D. W. Schaefer, J. E. Martin, P. Wiltzius, and D. Cannell, Phys.Rev. Lett., XX, XXX (1984)

8. T. A. Witten, Jr. and L. M. Sander, Phys.Rev.Lett.,fl,l400 (1981) 9. P. Meakin, Phys.Rev. ,A27, 604 (1983), 27, 2495 (1983), Phys.

Rev. Lett., g , 1119 (1983)

10. M. Kolb, R. Botet and J. Jullien, Phys.Rev.Lett.,z, 1123 (1983) 11. D. A. Weitz and M. Oliveira, Phys.Rev.Lett., XX, XXX (1984) 12. P. J. Flory, Principles of Polymer Chemistry (Cornell, Ithaca,

NY., 1953)

13. J. Isaacson and T. C. Lubensky, J. de Physique Lett.,O,L-469 (1980)

14. M. Daoud and J. F. Joanny, J. de Physique (Paris), G , ¹³⁵⁹

(1981)

15. M. Meakin, private communication

16. P. G. deGennes, Scaling Concepts in Polymer Physics, (Ithaca, Cornell, 1979)

17. P. G. deGennes, C. R. Acad.Sci.(Paris), 291, 17 (1980).