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Fractal-like structures in polystyrene solutions studied by light scattering intensity
~Solld S t a t e C o m m u n i c a t i o n s , Vol. 70, No. 3. pp. Printed in Great Britain.
0038-109818953.00+.00 Pergamon P r e s s plc
233-236, 1989.
FRACTAL-LIKE S T R U C T U R E S IN POLYSTYRENE SOLUTIONS STUDIED BY LIGHT SCATTERING INTENSITY S. M a g a z u ' , D. Majolino, F. Mallamace I s t i t u t o di Fisica d e l l ' U n i v e r s i t a ' di Meesina and G.N.S.M. - C.I.S.M 98166, Vill. S.Agata C.P. 55 (Messina) Italy and N. Micali, C. Vasi I s t i t u t o di T e e n i c h e S p e t t r o s c o p i c h e del C.N.R. 98166, Viii. S.Agata C.P. 55 (Messina) Italy ( R e c e i v e d 14 J u l y 1988 by R. F i e s c h i ) B y means o f a light s c a t t e r i n g e x p e r i m e n t as a f u n c t i o n of s c a t t e r e d w a v e v e c t o r , we h a v e d i r e c t l y s t u d i e d t h e aggregation p r o p e r t i e s o f p o l y s t y r e n e p a r t i c l e s in water s o l u t i o n s . F o r d i f f e r e n t ionic s t r e n g t h t h e s y s t e m s h o w s d i f f e r e n t k i n e t i c s with d i f f e r e n t f r a c t a l dimensions. At h i g h s t r e n g t h s a f a s t aggregation k i n e t i c s with a f r s c t a l dimension D = 1.75 ( d i f f u s i o n limited aggregation) is obtained, w h e r e a s a t low s t r e n g t h s t h e s y s t e m e v i d e n c e s a slow k i n e t i c s D ffi 2.1 (reaction limited aggregation).
q u a n t i f i e s t h e way in which mass M i n c r e a s e s with t h e i r l e n g t h R:
1.INTRODUCTION Aggregation p r o c e s s e s a r e o f c e n t r a l i n t e r e s t in m a n y f i e l d s o f science. Special e f f o r t h a s been made in t h e l a s t y e a r s t o u n d e r s t a n d t h e k i n e t i c p r o c e s s o f g r o w t h b y random aggregation, i.e. t h e mechanism o f c l u s t e r f o r m a t i o n f r o m small s u b u n i t s . In f a c t m a n y r e s u l t s c o n c e r n i n g t h e a g g r e g a t i o n o f gold, silica, p o l y s t y r e n e and biological colloids (1) a r e p r e s e n t in more r e c e n t l i t e r a t u r e .
M ~ RD As far as RLA is concerned, it is c h a r a c t e r i z e d b y a f r a c t a l dimension D ffi 2.1 =t=0.1 which has been directly observed in some e x p e r i m e n t s (4). The study of the density correlation f u n c t i o n f o r DLCCA b y c o m p u t e r s i m u l a t i o n (3) and e x p e r i m e n t a l r e s u l t s (4,6) g i v e a f r a c t a l dimension D = 1.75 4- 0.1. In s u c h a c a s e t h e same e x p o n e n t (6) c h a r a c t e r i z e s t h e i n c r e a s e with t h e time o f t h e c l u s t e r s i z e R ~ t lm.
S t a r t i n g f r o m t h e s i m p l e s t Eden model (2) (a l a t t i c e model in which p a r t i c l e s a r e added one a t a time, a t random times, to a d j a c e n t s i t e s in o r d e r to c l u s t e r ) s e v e r a l models h a v e been developed to explain t h e g r o w t h mechanism (1). Generally, t h e y can be c l a s s i f i e d i n t o two d i s t i n c t c l a s s e s : r e v e r s i b l e and i r r e v e r s i b l e aggregation; in t h e l a t t e r case, c o n t r a r i l y to t h e f o r m e r , t h e p a r t i c l e s a f t e r sticking together, cannot be separated into c o n s t i t u t i v e monomers. T h e r e v e r s i b l e aggregation is u s u a l l y called r e v e r s i b l e f l o c c u l a t i o n (RF). In t h e c a s e o f i r r e v e r s i b l e phenomena, we can d i s t i n g u i s h between a r e a c t i o n limited a g g r e g a t i o n (RLA) and a diffusion limited cluster-cluster aggregation (DLCCA).
P o l y s t y r e n e : w a t e r s o l u t i o n s are a very interesting system because they show beth m e c h a n i s m s o f aggregationl in f a c t , by changing t h e i r ionic s t r e n g t h t h e i r r e v e r s i b l e p r o c e s s e s become i r r e v e r s i b l e . Q u a s i - e l a s t i c light s c a t t e r i n g e x p e r i m e n t s show t h a t t h e s e colloidal s o l u t i o n s can h a v e f r a c t a l s t r u c t u r e (6) and r e a c h i r r e v e r s i b l e coagulation RLA p a s s i n g t h r o u g h a p h a s e of RF aggregation and s u b s e q u e n t l y t h r o u g h DLCCA aggregation. T h e e n t i r e p r o c e s s is ascribed to a gradual d e c r e a s i n g in t h e r e p u l s i v e p a r t of t h e pair potential. In f a c t t h e single particle h a s a n e t surface charge resulting from the partial d i s s o c i a t i o n o f p r o t o n s in t h e s u r f a c e c a r b o x y l g r o u p s . T h i s r e p u l s i v e p a r t o f t h e p o t e n t i a l can be shielded b y adding some s a l t (NaCI in o u r case). In p a r t i c u l a r , a t lower s a l t c o n t e n t , c~0.1mol/1, t h e ionic s t r e n g t h is v e r y low and t h e s y s t e m is in t h e r e v e r s i b l e regime RF; f o r a m o d e r a t e s a l t c o n t e s t , c>0.1mol/1, t h e s y s t e m goes into t h e i r r e v e r s i b l e aggregation regime. T h i s is c h a r a c t e r i z e d b y t h e p r e s e n c e o f two d i f f e r e n t kinetics, namely t h e RLA (0.1lmol/1); in t h e f i r s t c a s e t h e d y n a m i c s of t h e c l u s t e r s g r o w t h is slow while in t h e second one it is f a s t . However, the reported values for the ranges of c salt
T h e DLCCA can be explained as follows (3): a p a r t i c l e is placed a t t h e origin o f a lattice; a second one, placed a t a random site, at a given d i s t a n c e f r o m t h e origin, walks r a n d o m l y u n t i l it v i s i t s a s i t e a d j a c e n t to t h e f i r s t one. T h e n t h e walking p a r t i c l e becomes p a r t of a c l u s t e r and so on. T h i s c l u s t e r can d i f f u s e c o n t i n u i n g to grow f o r s u c c e s s i v e a g g r e g a t i o n s with o t h e r c l u s t e r s or single particles. In t h e RLA model t h e aggregation p r o c e s s s t a r t s when the constitutive monomer is somehow a c t i v a t e d (4). From a geometrical point o f view, t h e obtained c l u s t e r s are u s u a l l y c h a r a c t e r i z e d b y t h e i r f r a c t a l dimension D ( H a u s d o r f f dimension (5)) t h a t 233
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c o n c e n t r a t i o n s in w h i c h p o l y s t y r e n e p a r t i c l e s s h o w different aggregation mechanisms are strongly d e p e n d e n t on t h e r a d i u s o f t h e p a r t i c l e s (6). T h e aim o f t h i s w o r k is a d e t a i l e d s t u d y o f s u c h a g g r e g a t i o n p r o c e s s e s in p o l y s t y r e n e s o l u t i o n s by elastic light scattering that, unlike quasielastic s c a t t e r i n g , is a d i r e c t t e c h n i q u e . In f a c t , a s will be s h o w n in t h e n e x t s e c t i o n , s t r u c t u r a l p r o p e r t i e s a r e r e l a t e d t o i n t e n s i t y m e a s u r e m e n t s in a d i r e c t w a y , while dynamic measurements need the postulation of a p r e c i s e model. 2.LIGHT S C A T T E R I N G M E A S U R E M E N T S . S c a t t e r i n g is a p o w e r f u l tool f o r t h e s t u d y o f f r a c t a l s t r u c t u r e s ; d e p e n d i n g on t h e l e n g t h s c a l e o f i n t e r e s t a n d on t h e n a t u r e o f t h e a g g r e g a t e , i t is possible to use neutron, X ray and light scattering. In f a c t s i n c e t h e f r a c t a l o b j e c t s a r e s e l f - s i m i l a r structures (7,8) whose properties are scale i n v a r i a n t , t h e d i s t i n c t p a r t g(r) o f t h e i r pair c o r r e l a t i o n f u n c t i o n is h o m o g e n e o u s : (pCkrt)p(~r2))
= k-A/uoCrl)p(r2))
p(r) b e i n g t h e c o n c e n t a t i o n p o s i t i o n r; t h i s implies:
of
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d e p e n d on t h e c e n t e r - m a s s c l u s t e r s i n t e r n a l modes.
motion
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In t h e G u i n i e r r e g i m e (kR < < 1), f o r n e a r l y m o n o d i s p e r s e c o m p a c t a n d non i n t e r a c t i n g c l u s t e r s ( d i l u t e d p o l y m e r i c s o l u t i o n s (12)), t h e l i n e w i d t h is g i v e n b y ( r ) = DMk2, D M being t h e m u t u a l d i f f u s i o n c o e f f i c i e n t a n d D M ~ 1/E w h e r e ~ is t h e r a d i u s o f g y r a t i o n o f t h e c l u s t e r s . In a r e a l i s t i c c a s e t h e situation is more complicated (13) because interactions and polydispersity give a non e x p o n e n t i a l c o r r e l a t i o n f u n c t i o n (13,14). As a r e s u l t t h e r e is no d i r e c t r e l a t i o n b e t w e e n t h e mean r e l a x a t i o n r a t e and t h e c l u s t e r r a d i u s ~. In t h e P o r e d r e g i m e s c a l i n g a r g u m e n t s (I0) g i v e t h e f o l l o w i n g k d e p e n d e n c e o f t h e mean linewidth: ( r ) ~ k 3 ; kR > > 1 w h i c h p r o v e s to be i n d e p e n d e n t o f R. Such arguments indicate that intensity measurements as a function of the exchanged w a v e v e c t o r (eq.l,2) in t h e P o r e d regime p r o v i d e a d i r e c t e v a l u a t i o n o f t h e f r a c t a l d i m e n s i o n D. T h i s is a m o r e s u i t a b l e p r o c e d u r e r e s p e c t to t h e i n d i r e c t and not always possible measurement of the g y r a t i o n r a d i u s E in t e r m s o f t h e f i r s t c u m u l a n t in t h e l i g h t c o r r e l a t i o n f u n c t i o n in t h e G u i n i e r regime.
l0 s -
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N(R)=/ddr(p(O)p(r))/
~
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F o r a g e n e r i c c l u s t e r o f m a s s M i t is s h o w n (9) t h a t t h e s t r u c t u r e f a c t o r S(k), t h a t is j u s t t h e F o u r i e r t r a n s f o r m o f g(r), s a t i s f i e s t h e f o l l o w i n g g e n e r a l expression:
S~(k) = S(kR)
103 ~10 s
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This simple relationship gives the experimental conditions for the measurement of the fractal d i m e n s i o n D; in f a c t in t h e s o - c a l l e d P o r e d r e g i m e kR > > 1 (small d i s t a n c e s c o m p a r e d to t h e g y r a t i o n radius of the cluster and large compared to the chemical length l characterizing the monomer size) t h e m e a s u r e d i n t e n s i t y is: I(k) ~ S(k) ~ k -°
10 10~ _
b)
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(2)
The Pored scattering regime contrasts with the G u i n i e r s c a t t e r i n g r e g i m e (kR < < 1) w h e r e a G a u s s i a n d e c a y is e x p e c t e d . L i n e w i d t h m e a s u r e m e n t s c a n be o b t a i n e d f r o m t h e initial s l o p e o f a p h o t o n c o r r e l a t i o n f u n c t i o n that can give us information about the dynamics of s u c h f r a c t a l c l u s t e r s (10). T h e initial s l o p e ( f i r s t cumulant) is the mean relaxation rate (r) of spatial f l u c t u a t i o n s o f f r e q u e n c y k=47~k-tsin(0/2), w h e r e 0 is t h e s c a t t e r i n g a n g l e a n d k t h e w a v e l e n g t h in t h e m e d i u m . D e p e n d i n g on kR, t h e m e a s u r e d (F) m a y
10
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5 I0 k (pro -11
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Fig.1 I n t e n s i t y p r o f i l e s logI v s . logk f o r d i f f e r e n t s a l t c o n c e n t r a t i o n s . C i r c l e s a r e r e f e r r e d to t h e v a l u e s o b t a i n e d a t long t i m e s (at c o m p l e t e t h e a g g r e g a t i o n ) ; t r i a n g l e s a r e r e f e r r e d to i n t e r m e d i a t e times.
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~/1~
3 . E X P E R I M E N T A L D E T A I L S AND R E S U L T S . We used polystyrene latex particles, purchased f r o m t h e Dew C h e m i c a l Co., w i t h a d i a m e t e r o f R o ffi 0.085 ~ m a n d a s t a n d a r d d e v i a t i o n o f 0.0055~m. Before the measurement they were filtered and d i a l y s e d in o r d e r t o r e m o v e a n y c o n t a m i n a t i n g s u b s t a n c e . T h e m e a s u r e m e n t s w e r e m a d e in a w a t e r solution at a volume fraction of the dispersed phase o f r o u g h l y 5x10 -s. T h e a p p r o p r i a t e a m o u n t o f NaCI was a d d e d s o t h a t t h e a g g r e g a t i o n p r o c e s s c o u l d t a k e place. A t o u r c o n c e n t r a t i o n v a l u e s , m u l t i p l e scattering effects are not present as previously verified by attenuation and polarization checks. A c o m p u t e r c o n t r o l l e d g o n i o m e t e r w a s u s e d in a classical scattering geometry w i t h a 10roW unimode He-Ne laser operating at a wavelength of 6328 A a s i t s l i g h t s o u r c e . T h e d i m e n s i o n s o f t h e clusters, after the complete aggregation, depending on t h e a g g r e g a t i o n p r o c e s s , a r e in t h e 1 0 - - 3 0 t i m e s Ro range; these values are obtained by a quasielastic l i g h t s c a t t e r i n g (6) a n d v e r i f i e d b y u s . In o r d e r t o a s s u r e t h e P o r e d c o n d i t i o n s kR : ~ 1 a n a n g u l a r r a n g e 15 K 0 <:70 d e g r e e s w a s c h o s e n , c o r r e s p o n d i n g to t h e k r a n g e 3 . 3 7 # m - 1 ~ k ~ 1 4 . 8 1 # m -1. S u c h a k r a n g e is l i m i t e d t o s t u d y a f r a c t a l s y s t e m , u s u a l l y t h e r e g i o n o f s e l f s i m i l a r i t y in s u c h s y s t e m s is o b s e r v e d on a l a r g e r s c a l e o f l e n g t h s , b u t it is enough to point out the existence of a scaling behaviour. The salt concentration explored is 0.015<:c~3 m o l / l t h a t s p a n s in all t h e a g g r e g a t i o n r e g i m e s . A n y c o n c e n t r a t i o n is f o l l o w e d in t i m e u n t i l i t s c o m p l e t e a g g r e g a t i o n ; in t h e c a s e o f R F k i n e t i c s t h e c o m p l e t e a g g r e g a t i o n is Yeached a f t e r s o m e days. The collected intensity data have been corrected for the well-known dependence of the s c a t t e r i n g v o l u m e b y t h e s c a t t e r i n g angle. In fig. 1 we r e p o r t s o m e t y p i c a l i n t e n s i t y p r o f i l e s f o r t h e m o l / l c o n c e n t r a t i o n s o f cffi0.075 (a), 0.15 (b) and 1.5 (c) a f t e r t h e s y s t e m h a s r e a c h e d a stationary condition (full circles) and for intermediate times (triangles); the plots are representative of the different mechanisms of a g g r e g a t i o n R F (a), RLA (b) a n d DLCCA (c). T h e continuous lines are the best fits of the e x p e r i m e n t a l r e s u l t s b y eq.2. In fig.2 t h e m e a s u r e d s l o p e (log I v s . log k) of the obtained intensity profiles at different times is r e p o r t e d f o r t h e s a l t c o n c e n t r a t i o n s c= 0.075, 0.15, 1.5, 3 mol/l. A s c a n be s e e n t h e c o m p l e t e a g g r e g a t i o n is o b t a i n e d a t d i f f e r e n t times for d i f f e r e n t c o n c e n t r a t i o n s i.e. a t h i g h c o n c e n t r a t i o n s t h e k i n e t i c r e g i m e is t h e DLCCA (Dffil.75 ± 0.01) a n d o u r s is a f a s t p r o c e s s ; a t cffi0.15 m o l / l we a r e in slow r e g i m e a n d t h e o b t a i n e d f r a c t a l d i m e n s i o n (Dffi2.1 -t- 0.01) i n d i c a t e s a n RLA a g g r e g a t i o n . F o r c=0.075mol/1 t h e a g g r e g a t i o n p r o c e s s is v e r y slow ( s o m e d a y s ) , we a r e in p r e s e n c e o f r e v e r s i b l e f l o c c u l a t i o n (RF) a n d t h e s y s t e m e x h i b i t s e v i d e n c e that complete aggregation is o b t a i n e d passing t h r o u g h a n i n t e r m e d i a t e s t a t e . T h i s p i c t u r e is a l s o c o n f i r m e d a t c=0.05mol/1. T h e o r i g i n o f s u c h a b e h a v i o u r is u p to now u n c l e a r a n d m o r e a c c u r a t e m e a s u r e m e n t s a r e in p r o g r e s s to c l a r i f y t h i s point. T h e t i m e b e h a v i o u r o f t h e f a s t r e g i m e is in c l o s e agreement with the numeric solutions of the S m o l u c h o w s k i e q u a t i o n f o r DLCCA a g g r e g a t i o n in o u r s y s t e m (6). F i n a l l y , in fig.3 we r e p o r t t h e o b t a i n e d t i m e s T f o r complete aggregation as a function of the salt concentration. This result agrees with the Cametti
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Fig.2 M e a s u r e d s l o p e s o f t h e i n t e n s i t y a t d i f f e r e n t t i m e s f o r t h e s a l t c o n c e n t r a t i o n s (NaCI) c=0.075; 0.15;1.5 and 3 mol/l.
RF
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Cmot / t Fig.3 T i m e s T f o r c o m p l e t e a g g r e g a t i o n f u n c t i o n o f t h e s a l t c o n c e n t r a t i o n c.
as
236
FRACTAL-LIKE
et al. (6) calculation rela{ed to t h e shape o f t h e interaction potential between two monomers, where t h e aggregation r a t e is d i r e c t l y connected t o t h e ionic s t r e n g t h of t h e solution. T h e obtained b e h a v i o u r o f ~- (the i n v e r s e r a t e o f passage above t h e potential barrier) vs. t h e salt concentration c is similar to t h a t of our complete aggregation time T. In fig. 3, we also r e p o r t t h e corresponding time regions o b s e r v e d f o r t h e d i f f e r e n t aggregation processes. T h e s e r e s u l t s are also in agreement with t h e cited (6) light beating measurements . In conclusion we h a v e r e p o r t e d a d i r e c t measurement (by a scaling b e h a v i o u r o b s e r v e d in a
STRUCTURES
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limited k range) o f t h e d i f f e r e n t aggregation processes in p o l y s t y r e n e particles as a f u n c t i o n o f t h e ionic s t r e n g t h , namely t h e u l t r a - s l o w r e v e r s i b l e flocculation RF, the slow reaction limited aggregation RLA and t h e f a s t diffusion limited aggregation DLCCA. Our r e s u l t s are in agreement with previous dynamical light scattering measurements (6). Acknowledgments. T h e a u t h o r s would like to thank Prof. P. Tartaglia f o r helpful discussion. S.Interdonato is also acknowledged for t h e p r o j e c t and installation of t h e electronic hardware of t h e goniometer.
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8. 9. 10. 11.
12. 13.
R. Richterl, L.M. Sander and Z. Cheng, J. Colloid Interf. Sci. ~ 203 (1984). J.E.Martin and BJ.Ackerson, Phys.Rev.A 3_1_, 1180 (1985). D.W. S c h a e f e r and C.C.Han in Dynamic L ~ h t Scatterin~ edited by R.Pecora (Plenum, New York 1985). W.Hess in Light Scatterina in Liquids and affacromolecular Solutions edited by V.Degiorgio, M.Corti and M.Giglio (Plenum, New York 1980). P.N.Pusey and R.J.A. Tough in Dynamic Light S c a t t e r / n ~ edited by R.Pecora (Plenum, New York 1985). P.N.Pusey, H.M.Fijnault and A.Vrij J.Chem. P h y s . ~ 4270 (1982).
0038-109818953.00+.00 Pergamon P r e s s plc
233-236, 1989.
FRACTAL-LIKE S T R U C T U R E S IN POLYSTYRENE SOLUTIONS STUDIED BY LIGHT SCATTERING INTENSITY S. M a g a z u ' , D. Majolino, F. Mallamace I s t i t u t o di Fisica d e l l ' U n i v e r s i t a ' di Meesina and G.N.S.M. - C.I.S.M 98166, Vill. S.Agata C.P. 55 (Messina) Italy and N. Micali, C. Vasi I s t i t u t o di T e e n i c h e S p e t t r o s c o p i c h e del C.N.R. 98166, Viii. S.Agata C.P. 55 (Messina) Italy ( R e c e i v e d 14 J u l y 1988 by R. F i e s c h i ) B y means o f a light s c a t t e r i n g e x p e r i m e n t as a f u n c t i o n of s c a t t e r e d w a v e v e c t o r , we h a v e d i r e c t l y s t u d i e d t h e aggregation p r o p e r t i e s o f p o l y s t y r e n e p a r t i c l e s in water s o l u t i o n s . F o r d i f f e r e n t ionic s t r e n g t h t h e s y s t e m s h o w s d i f f e r e n t k i n e t i c s with d i f f e r e n t f r a c t a l dimensions. At h i g h s t r e n g t h s a f a s t aggregation k i n e t i c s with a f r s c t a l dimension D = 1.75 ( d i f f u s i o n limited aggregation) is obtained, w h e r e a s a t low s t r e n g t h s t h e s y s t e m e v i d e n c e s a slow k i n e t i c s D ffi 2.1 (reaction limited aggregation).
q u a n t i f i e s t h e way in which mass M i n c r e a s e s with t h e i r l e n g t h R:
1.INTRODUCTION Aggregation p r o c e s s e s a r e o f c e n t r a l i n t e r e s t in m a n y f i e l d s o f science. Special e f f o r t h a s been made in t h e l a s t y e a r s t o u n d e r s t a n d t h e k i n e t i c p r o c e s s o f g r o w t h b y random aggregation, i.e. t h e mechanism o f c l u s t e r f o r m a t i o n f r o m small s u b u n i t s . In f a c t m a n y r e s u l t s c o n c e r n i n g t h e a g g r e g a t i o n o f gold, silica, p o l y s t y r e n e and biological colloids (1) a r e p r e s e n t in more r e c e n t l i t e r a t u r e .
M ~ RD As far as RLA is concerned, it is c h a r a c t e r i z e d b y a f r a c t a l dimension D ffi 2.1 =t=0.1 which has been directly observed in some e x p e r i m e n t s (4). The study of the density correlation f u n c t i o n f o r DLCCA b y c o m p u t e r s i m u l a t i o n (3) and e x p e r i m e n t a l r e s u l t s (4,6) g i v e a f r a c t a l dimension D = 1.75 4- 0.1. In s u c h a c a s e t h e same e x p o n e n t (6) c h a r a c t e r i z e s t h e i n c r e a s e with t h e time o f t h e c l u s t e r s i z e R ~ t lm.
S t a r t i n g f r o m t h e s i m p l e s t Eden model (2) (a l a t t i c e model in which p a r t i c l e s a r e added one a t a time, a t random times, to a d j a c e n t s i t e s in o r d e r to c l u s t e r ) s e v e r a l models h a v e been developed to explain t h e g r o w t h mechanism (1). Generally, t h e y can be c l a s s i f i e d i n t o two d i s t i n c t c l a s s e s : r e v e r s i b l e and i r r e v e r s i b l e aggregation; in t h e l a t t e r case, c o n t r a r i l y to t h e f o r m e r , t h e p a r t i c l e s a f t e r sticking together, cannot be separated into c o n s t i t u t i v e monomers. T h e r e v e r s i b l e aggregation is u s u a l l y called r e v e r s i b l e f l o c c u l a t i o n (RF). In t h e c a s e o f i r r e v e r s i b l e phenomena, we can d i s t i n g u i s h between a r e a c t i o n limited a g g r e g a t i o n (RLA) and a diffusion limited cluster-cluster aggregation (DLCCA).
P o l y s t y r e n e : w a t e r s o l u t i o n s are a very interesting system because they show beth m e c h a n i s m s o f aggregationl in f a c t , by changing t h e i r ionic s t r e n g t h t h e i r r e v e r s i b l e p r o c e s s e s become i r r e v e r s i b l e . Q u a s i - e l a s t i c light s c a t t e r i n g e x p e r i m e n t s show t h a t t h e s e colloidal s o l u t i o n s can h a v e f r a c t a l s t r u c t u r e (6) and r e a c h i r r e v e r s i b l e coagulation RLA p a s s i n g t h r o u g h a p h a s e of RF aggregation and s u b s e q u e n t l y t h r o u g h DLCCA aggregation. T h e e n t i r e p r o c e s s is ascribed to a gradual d e c r e a s i n g in t h e r e p u l s i v e p a r t of t h e pair potential. In f a c t t h e single particle h a s a n e t surface charge resulting from the partial d i s s o c i a t i o n o f p r o t o n s in t h e s u r f a c e c a r b o x y l g r o u p s . T h i s r e p u l s i v e p a r t o f t h e p o t e n t i a l can be shielded b y adding some s a l t (NaCI in o u r case). In p a r t i c u l a r , a t lower s a l t c o n t e n t , c~0.1mol/1, t h e ionic s t r e n g t h is v e r y low and t h e s y s t e m is in t h e r e v e r s i b l e regime RF; f o r a m o d e r a t e s a l t c o n t e s t , c>0.1mol/1, t h e s y s t e m goes into t h e i r r e v e r s i b l e aggregation regime. T h i s is c h a r a c t e r i z e d b y t h e p r e s e n c e o f two d i f f e r e n t kinetics, namely t h e RLA (0.1
T h e DLCCA can be explained as follows (3): a p a r t i c l e is placed a t t h e origin o f a lattice; a second one, placed a t a random site, at a given d i s t a n c e f r o m t h e origin, walks r a n d o m l y u n t i l it v i s i t s a s i t e a d j a c e n t to t h e f i r s t one. T h e n t h e walking p a r t i c l e becomes p a r t of a c l u s t e r and so on. T h i s c l u s t e r can d i f f u s e c o n t i n u i n g to grow f o r s u c c e s s i v e a g g r e g a t i o n s with o t h e r c l u s t e r s or single particles. In t h e RLA model t h e aggregation p r o c e s s s t a r t s when the constitutive monomer is somehow a c t i v a t e d (4). From a geometrical point o f view, t h e obtained c l u s t e r s are u s u a l l y c h a r a c t e r i z e d b y t h e i r f r a c t a l dimension D ( H a u s d o r f f dimension (5)) t h a t 233
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c o n c e n t r a t i o n s in w h i c h p o l y s t y r e n e p a r t i c l e s s h o w different aggregation mechanisms are strongly d e p e n d e n t on t h e r a d i u s o f t h e p a r t i c l e s (6). T h e aim o f t h i s w o r k is a d e t a i l e d s t u d y o f s u c h a g g r e g a t i o n p r o c e s s e s in p o l y s t y r e n e s o l u t i o n s by elastic light scattering that, unlike quasielastic s c a t t e r i n g , is a d i r e c t t e c h n i q u e . In f a c t , a s will be s h o w n in t h e n e x t s e c t i o n , s t r u c t u r a l p r o p e r t i e s a r e r e l a t e d t o i n t e n s i t y m e a s u r e m e n t s in a d i r e c t w a y , while dynamic measurements need the postulation of a p r e c i s e model. 2.LIGHT S C A T T E R I N G M E A S U R E M E N T S . S c a t t e r i n g is a p o w e r f u l tool f o r t h e s t u d y o f f r a c t a l s t r u c t u r e s ; d e p e n d i n g on t h e l e n g t h s c a l e o f i n t e r e s t a n d on t h e n a t u r e o f t h e a g g r e g a t e , i t is possible to use neutron, X ray and light scattering. In f a c t s i n c e t h e f r a c t a l o b j e c t s a r e s e l f - s i m i l a r structures (7,8) whose properties are scale i n v a r i a n t , t h e d i s t i n c t p a r t g(r) o f t h e i r pair c o r r e l a t i o n f u n c t i o n is h o m o g e n e o u s : (pCkrt)p(~r2))
= k-A/uoCrl)p(r2))
p(r) b e i n g t h e c o n c e n t a t i o n p o s i t i o n r; t h i s implies:
of
monomers
at
the
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d e p e n d on t h e c e n t e r - m a s s c l u s t e r s i n t e r n a l modes.
motion
or
on
the
In t h e G u i n i e r r e g i m e (kR < < 1), f o r n e a r l y m o n o d i s p e r s e c o m p a c t a n d non i n t e r a c t i n g c l u s t e r s ( d i l u t e d p o l y m e r i c s o l u t i o n s (12)), t h e l i n e w i d t h is g i v e n b y ( r ) = DMk2, D M being t h e m u t u a l d i f f u s i o n c o e f f i c i e n t a n d D M ~ 1/E w h e r e ~ is t h e r a d i u s o f g y r a t i o n o f t h e c l u s t e r s . In a r e a l i s t i c c a s e t h e situation is more complicated (13) because interactions and polydispersity give a non e x p o n e n t i a l c o r r e l a t i o n f u n c t i o n (13,14). As a r e s u l t t h e r e is no d i r e c t r e l a t i o n b e t w e e n t h e mean r e l a x a t i o n r a t e and t h e c l u s t e r r a d i u s ~. In t h e P o r e d r e g i m e s c a l i n g a r g u m e n t s (I0) g i v e t h e f o l l o w i n g k d e p e n d e n c e o f t h e mean linewidth: ( r ) ~ k 3 ; kR > > 1 w h i c h p r o v e s to be i n d e p e n d e n t o f R. Such arguments indicate that intensity measurements as a function of the exchanged w a v e v e c t o r (eq.l,2) in t h e P o r e d regime p r o v i d e a d i r e c t e v a l u a t i o n o f t h e f r a c t a l d i m e n s i o n D. T h i s is a m o r e s u i t a b l e p r o c e d u r e r e s p e c t to t h e i n d i r e c t and not always possible measurement of the g y r a t i o n r a d i u s E in t e r m s o f t h e f i r s t c u m u l a n t in t h e l i g h t c o r r e l a t i o n f u n c t i o n in t h e G u i n i e r regime.
l0 s -
R
N(R)=/ddr(p(O)p(r))/
~
Rd-A
=
R D
10 ~
J o
F o r a g e n e r i c c l u s t e r o f m a s s M i t is s h o w n (9) t h a t t h e s t r u c t u r e f a c t o r S(k), t h a t is j u s t t h e F o u r i e r t r a n s f o r m o f g(r), s a t i s f i e s t h e f o l l o w i n g g e n e r a l expression:
S~(k) = S(kR)
103 ~10 s
a) I
I
I
e-
a n d in a d - d i m e n s i o n a l e x p a n s i o n , .d k2R 2 1 - T S(kR)
for
~I0 ~
kR << 1 (1)
=
(kR) -v
for
kR > > I
This simple relationship gives the experimental conditions for the measurement of the fractal d i m e n s i o n D; in f a c t in t h e s o - c a l l e d P o r e d r e g i m e kR > > 1 (small d i s t a n c e s c o m p a r e d to t h e g y r a t i o n radius of the cluster and large compared to the chemical length l characterizing the monomer size) t h e m e a s u r e d i n t e n s i t y is: I(k) ~ S(k) ~ k -°
10 10~ _
b)
I
I
I
I
I
10~
(2)
The Pored scattering regime contrasts with the G u i n i e r s c a t t e r i n g r e g i m e (kR < < 1) w h e r e a G a u s s i a n d e c a y is e x p e c t e d . L i n e w i d t h m e a s u r e m e n t s c a n be o b t a i n e d f r o m t h e initial s l o p e o f a p h o t o n c o r r e l a t i o n f u n c t i o n that can give us information about the dynamics of s u c h f r a c t a l c l u s t e r s (10). T h e initial s l o p e ( f i r s t cumulant) is the mean relaxation rate (r) of spatial f l u c t u a t i o n s o f f r e q u e n c y k=47~k-tsin(0/2), w h e r e 0 is t h e s c a t t e r i n g a n g l e a n d k t h e w a v e l e n g t h in t h e m e d i u m . D e p e n d i n g on kR, t h e m e a s u r e d (F) m a y
10
c)
2
5 I0 k (pro -11
I
15
Fig.1 I n t e n s i t y p r o f i l e s logI v s . logk f o r d i f f e r e n t s a l t c o n c e n t r a t i o n s . C i r c l e s a r e r e f e r r e d to t h e v a l u e s o b t a i n e d a t long t i m e s (at c o m p l e t e t h e a g g r e g a t i o n ) ; t r i a n g l e s a r e r e f e r r e d to i n t e r m e d i a t e times.
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FRACTAL-LIKE STRUCTURES
~/1~
3 . E X P E R I M E N T A L D E T A I L S AND R E S U L T S . We used polystyrene latex particles, purchased f r o m t h e Dew C h e m i c a l Co., w i t h a d i a m e t e r o f R o ffi 0.085 ~ m a n d a s t a n d a r d d e v i a t i o n o f 0.0055~m. Before the measurement they were filtered and d i a l y s e d in o r d e r t o r e m o v e a n y c o n t a m i n a t i n g s u b s t a n c e . T h e m e a s u r e m e n t s w e r e m a d e in a w a t e r solution at a volume fraction of the dispersed phase o f r o u g h l y 5x10 -s. T h e a p p r o p r i a t e a m o u n t o f NaCI was a d d e d s o t h a t t h e a g g r e g a t i o n p r o c e s s c o u l d t a k e place. A t o u r c o n c e n t r a t i o n v a l u e s , m u l t i p l e scattering effects are not present as previously verified by attenuation and polarization checks. A c o m p u t e r c o n t r o l l e d g o n i o m e t e r w a s u s e d in a classical scattering geometry w i t h a 10roW unimode He-Ne laser operating at a wavelength of 6328 A a s i t s l i g h t s o u r c e . T h e d i m e n s i o n s o f t h e clusters, after the complete aggregation, depending on t h e a g g r e g a t i o n p r o c e s s , a r e in t h e 1 0 - - 3 0 t i m e s Ro range; these values are obtained by a quasielastic l i g h t s c a t t e r i n g (6) a n d v e r i f i e d b y u s . In o r d e r t o a s s u r e t h e P o r e d c o n d i t i o n s kR : ~ 1 a n a n g u l a r r a n g e 15 K 0 <:70 d e g r e e s w a s c h o s e n , c o r r e s p o n d i n g to t h e k r a n g e 3 . 3 7 # m - 1 ~ k ~ 1 4 . 8 1 # m -1. S u c h a k r a n g e is l i m i t e d t o s t u d y a f r a c t a l s y s t e m , u s u a l l y t h e r e g i o n o f s e l f s i m i l a r i t y in s u c h s y s t e m s is o b s e r v e d on a l a r g e r s c a l e o f l e n g t h s , b u t it is enough to point out the existence of a scaling behaviour. The salt concentration explored is 0.015<:c~3 m o l / l t h a t s p a n s in all t h e a g g r e g a t i o n r e g i m e s . A n y c o n c e n t r a t i o n is f o l l o w e d in t i m e u n t i l i t s c o m p l e t e a g g r e g a t i o n ; in t h e c a s e o f R F k i n e t i c s t h e c o m p l e t e a g g r e g a t i o n is Yeached a f t e r s o m e days. The collected intensity data have been corrected for the well-known dependence of the s c a t t e r i n g v o l u m e b y t h e s c a t t e r i n g angle. In fig. 1 we r e p o r t s o m e t y p i c a l i n t e n s i t y p r o f i l e s f o r t h e m o l / l c o n c e n t r a t i o n s o f cffi0.075 (a), 0.15 (b) and 1.5 (c) a f t e r t h e s y s t e m h a s r e a c h e d a stationary condition (full circles) and for intermediate times (triangles); the plots are representative of the different mechanisms of a g g r e g a t i o n R F (a), RLA (b) a n d DLCCA (c). T h e continuous lines are the best fits of the e x p e r i m e n t a l r e s u l t s b y eq.2. In fig.2 t h e m e a s u r e d s l o p e (log I v s . log k) of the obtained intensity profiles at different times is r e p o r t e d f o r t h e s a l t c o n c e n t r a t i o n s c= 0.075, 0.15, 1.5, 3 mol/l. A s c a n be s e e n t h e c o m p l e t e a g g r e g a t i o n is o b t a i n e d a t d i f f e r e n t times for d i f f e r e n t c o n c e n t r a t i o n s i.e. a t h i g h c o n c e n t r a t i o n s t h e k i n e t i c r e g i m e is t h e DLCCA (Dffil.75 ± 0.01) a n d o u r s is a f a s t p r o c e s s ; a t cffi0.15 m o l / l we a r e in slow r e g i m e a n d t h e o b t a i n e d f r a c t a l d i m e n s i o n (Dffi2.1 -t- 0.01) i n d i c a t e s a n RLA a g g r e g a t i o n . F o r c=0.075mol/1 t h e a g g r e g a t i o n p r o c e s s is v e r y slow ( s o m e d a y s ) , we a r e in p r e s e n c e o f r e v e r s i b l e f l o c c u l a t i o n (RF) a n d t h e s y s t e m e x h i b i t s e v i d e n c e that complete aggregation is o b t a i n e d passing t h r o u g h a n i n t e r m e d i a t e s t a t e . T h i s p i c t u r e is a l s o c o n f i r m e d a t c=0.05mol/1. T h e o r i g i n o f s u c h a b e h a v i o u r is u p to now u n c l e a r a n d m o r e a c c u r a t e m e a s u r e m e n t s a r e in p r o g r e s s to c l a r i f y t h i s point. T h e t i m e b e h a v i o u r o f t h e f a s t r e g i m e is in c l o s e agreement with the numeric solutions of the S m o l u c h o w s k i e q u a t i o n f o r DLCCA a g g r e g a t i o n in o u r s y s t e m (6). F i n a l l y , in fig.3 we r e p o r t t h e o b t a i n e d t i m e s T f o r complete aggregation as a function of the salt concentration. This result agrees with the Cametti
2
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-
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~
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Fig.2 M e a s u r e d s l o p e s o f t h e i n t e n s i t y a t d i f f e r e n t t i m e s f o r t h e s a l t c o n c e n t r a t i o n s (NaCI) c=0.075; 0.15;1.5 and 3 mol/l.
RF
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% e~ 10 ~ _
",%
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103 0
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I 2
I 3
Cmot / t Fig.3 T i m e s T f o r c o m p l e t e a g g r e g a t i o n f u n c t i o n o f t h e s a l t c o n c e n t r a t i o n c.
as
236
FRACTAL-LIKE
et al. (6) calculation rela{ed to t h e shape o f t h e interaction potential between two monomers, where t h e aggregation r a t e is d i r e c t l y connected t o t h e ionic s t r e n g t h of t h e solution. T h e obtained b e h a v i o u r o f ~- (the i n v e r s e r a t e o f passage above t h e potential barrier) vs. t h e salt concentration c is similar to t h a t of our complete aggregation time T. In fig. 3, we also r e p o r t t h e corresponding time regions o b s e r v e d f o r t h e d i f f e r e n t aggregation processes. T h e s e r e s u l t s are also in agreement with t h e cited (6) light beating measurements . In conclusion we h a v e r e p o r t e d a d i r e c t measurement (by a scaling b e h a v i o u r o b s e r v e d in a
STRUCTURES
Vol. 70, No. 3
limited k range) o f t h e d i f f e r e n t aggregation processes in p o l y s t y r e n e particles as a f u n c t i o n o f t h e ionic s t r e n g t h , namely t h e u l t r a - s l o w r e v e r s i b l e flocculation RF, the slow reaction limited aggregation RLA and t h e f a s t diffusion limited aggregation DLCCA. Our r e s u l t s are in agreement with previous dynamical light scattering measurements (6). Acknowledgments. T h e a u t h o r s would like to thank Prof. P. Tartaglia f o r helpful discussion. S.Interdonato is also acknowledged for t h e p r o j e c t and installation of t h e electronic hardware of t h e goniometer.
REFERENCES
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