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CLUSTER AGGREGATION
FRACTALS IN PHYSICS L. Pietronero, E. Tosatti (editors) © Elsevier Science Publishers Β. V., 1986
255
CLUSTER AGGREGATION
R. BOTET, R. J U L L I E N and M. KOLB Laboratoire France
de P h y s i q u e de$. S o l i d e s , Ba*t. 510,
Universite
P a r i s - S u d , C e n t r e d ' O r s a y , 91405 O r s a y ,
We i n t r o d u c e the c l u s t e r i n g o f c l u s t e r s p r o c e s s a s a model to d e s c r i b e a g g r e g a t i o n o f c o l l o i d a l o r a e r o s o l p a r t i c l e s i n the low c o n c e n t r a t i o n r e g i m e . We i n v e s t i g a t e m o d i f i c a t i o n s o f the parameters of the model. I n p a r t i c u l a r , v a r i o u s r e v e r s i b l e models o f a g g r e g a t i o n are d e t a i l e d and we d i s c u s s p o s s i b l e r e l a t i o n s between them ( u n i v e r s a l i t y c l a s s e s ) . 1 . INTRODUCTION
an i n f i n i t e medium, s t a t i s t i c a l l y homogeneous,
The f o r m a t i o n o f a g g r e g a t e s by c l u s t e r i n g isolated particles
p l a y s an i m p o r t a n t
numerous s c i e n t i f i c sols,
1
areas .
well-defined
in
been found u s i n g numerical
brownian
of monodisperse
particles.
they are on two n e i g h b o u r i n g s i t e s o f the .
e x p l a n a t i o n has
they s t i c k i r r e v e r s i b l y dimer.
s i m u l a t i o n s . We do
know why t h e s e a g g r e g a t e s are
d e n s i t y N Q/ L
When two o f t h e s e p a r t i c l e s c o l l i d e ( i . e . when
geometrical 2 3
on many l e n g t h - s c a l e s '
A beginning for a theoretical
not y e t
w i t h an i n i t i a l
In c o l l o i d s a n d aero
a g g r e g a t e s show a s e l f - s i m i l a r
structure,
role
3
of
which i s a
function
o f i t s m a s s . When two c l u s t e r s c o l l i d e ,
but we b e g i n to know how they are f r a c t a l
and
s t i c k and form a l a r g e r r i g i d c l u s t e r ,
how t h e i r f r a c t a l
physics.
on. Since s t i c k i n g i s i r r e v e r s i b l e ,
d i m e n s i o n i s r e l a t e d to
I n the f o l l o w i n g , we i n t r o d u c e irreversible
cluster-cluster
which g i v e s a r e a l i s t i c
the model
of
aggregation ( C I - C I )
d e s c r i p t i o n of
colloid
rigid
T h i s c l u s t e r a l s o f o l l o w s a random walk
trajectory with a v e l o c i t y
fractal
lattice)
and form a s m a l l
they and so
the
number
o f c l u s t e r s d e c r e a s e s w i t h time and the p r o c e s s ends when t h e r e remains o n l y one s i n g l e c l u s t e r i n the box.
and a e r o s o l a g g r e g a t i o n .
I n t h i s v e r s i o n , the u n d e r l y i n g l a t t i c e
for
2. THE MODEL OF CLUSTERING OF CLUSTERS
b i d s c o n t i n u o u s r o t a t i o n o f the c l u s t e r s a l o n g
T h i s model has been proposed i n d e p e n d e n t l y by 4 5 Meakin and by us . We d e s c r i b e here o n l y the
their trajectory.
3-dimensional
s i m u l a t i o n s have been done w i t h a
v e r s i o n o f t h i s model s i n c e i t
the most u s e f u l riments
for direct
. Nevertheless all
6 have been n u m e r i c a l l y
is
comparison w i t h expedimensions l e s s
than
s t u d i e d ^ as well as d i
mensions l a r g e r than the upper c r i t i c a l sion Log as f o l l o w s
I
The model
: i n a box L χ L χ L w i t h
is
have not been a l l o w e d to r o t a t e .
defined
tion
generalization
L i k e w i s e , the r e l a t i o n
between v e l o c i t y
mass i s unimportant
f o r the g e o m e t r i c a l
if
that .is i f
it
is realistic,
periodic
the v e l o c i t y
not an i n c r e a s i n g f u n c t i o n o f the m a s s ^ .
i n f i n i t e c u b i c l a t t i c e , we put randomly N Q i d e n
a f u n c t i o n o f time depends
Each o f t h e s e p a r t i c l e s
a random walk on the l a t t i c e . The model
follows simulates
F i g u r e 1 shows a t y p i c a l
is
Of
o f the c l u s t e r s as
on t h i s
relation
c l u s t e r o f 1024
t i c l e s , grown by t h i s p r o c e s s i n a box o f 70 χ 70 χ 70.
and
features, 1
boundary c o n d i t i o n s , which i s a p o r t i o n o f an
particles.
Off-lattice
i s reasonable .
c o u r s e , the s i z e d i s t r i b u t i o n
tical
clusters
o f the random w a l k , and they show t h a t c l u s t e r s have the same f r a c t a l s t r u c t u r e i f t h i s r o t a g
dimen-
Log36
o
I n the s i m u l a t i o n the
1 1
.
par
size
R. Botet et al.
256
When a v e r a g i n g o v e r a l a r g e number o f
clus
t e r s , Log-Log p l o t of radius of g y r a t i o n versus mass shows a s c a l i n g r e l a t i o n
o f the form :
radii.
length scales greater D i s the f r a c t a l
i s D = 1.78
than a few monomer
d i m e n s i o n and i t s
value
± 0.05.
the experimental
MODIFYING THE STARTING CONDITIONS I f we a l l o w p o l y d i s p e r s i t y o f the monomers
( b a l l s with a d i s t r i b u t i o n of r a d i i )
the same
k i n d o f s i m u l a t i o n s show t h a t the r e s u l t i n g a g g r e
Ν ^ RD for all
4.
T h i s v a l u e i s v e r y c l o s e to ? v a l u e s o f F o r r e s t and W i t t e n
g a t e s are s t i l l
fractal
o b j e c t s , w i t h the same 12
f r a c t a l d i m e n s i o n a s i n the monodisperse case These numerical r e s u l t s have been s u c c e s s f u l l y
.
compared w i t h r e c e n t experiments on p o l y d i s p e r s e 13 Fe a g g r e g a t e s . M o r e o v e r , as time goes o n , the effective density
:
I - 1 ^ R^"u i=all clusters L becomes o f o r d e r u n i t y , s i n c e the mean c l u s t e r r a d i u s R i n c r e a s e s w i t h t i m e . Then the screening i s i n e f f i c i e n t
because o f
We are i n the s o - c a l l e d k i n e t i c and the f r a c t a l me i s
brownian
entanglement.
g e l a t i o n regime
dimension t y p i c a l
of t h i s
regi
D = 1.75 ± 0 . 0 7 14 ( i n d = 2 ) .
5 . MODIFYING THE DIFFUSION CONDITIONS We can i m a g i n e a m o l e c u l a r ( i n s t e a d o f brow n i a n ) d i f f u s i o n o f the c l u s t e r s , where the mean FIGURE 1 T y p i c a l 3 - d i m e n s i o n a l c l u s t e r o f 1024 grown by C l - C l p r o c e s s on a l a t t i c e .
f r e e path i s o n l y l i m i t e d particles,
clusters.
quenched Fe v a p o r s , and o f
Weitz and O l i v e r i a 3 : D = 1.75
± 0.05
on g o l d
colloids.
The f r a c t a l
clusters
Here,
is interesting
t o note how few
parameters
are needed to r e c o v e r the experimental The s t a r t i n g c o n d i t i o n s a r e
results,
: monodisperse
p a r t i c l e s a t low c o n c e n t r a t i o n . The d i f f u s i o n c o n d i t i o n s are
: brownian
diffusion. The s t i c k i n g c o n d i t i o n s a r e
: o n c e two
t e r s c o l l i d e , they s t i c k i r r e v e r s i b l y new r i g i d
other.
more deeply i n t o each
d i m e n s i o n o f the
i s D = 1.91 ±
resulting
0 . 0 3 1 .5
6 . MODIFYING THE S T I C K I N G CONDITIONS
3 . PHYSICAL PARAMETERS OF THE MODEL It
effi
c i e n t than i n the brownian case and on average the c l u s t e r s p e n e t r a t e
D=1.8±0.1on
by c o l l i s i o n s between
Here a l s o the s c r e e n i n g i s l e s s
clus
and form a
cluster.
What happens when we change some o f the meters o f the model ?
t h r e e parameters can be m o d i f i e d :
p r o b a b i l i t y o f s t i c k i n g when c o l l i d i n g , the d i t y o f the c l u s t e r s and the i r r e v e r s i b i l i t y
of
the s t i c k i n g p r o c e s s . I f we l e t
the s t i c k i n g p r o b a b i l i t y
tend to
we o b t a i n the chemical model. I n t h i s c a s e ,
0,
two
c l u s t e r s must c o l l i d e many t i m e s b e f o r e s t i c k i n g . B u t once s t u c k , i r r e v e r s i b i l i t y
implies
that
c l u s t e r s can not b r e a k . T h i s chemical model has
1 been s t u d i e d n u m e r i c a l l y
para
the rigi
(the
fi and
17
experimentally
h e i g h t o f the r e p u l s i v e b a r r i e r
between
two g o l d c o l l o i d s can be v a r i e d c h e m i c a l l y ) . The fractal D = 2.00
d i m e n s i o n o f the r e s u l t i n g c l u s t e r s i s ± 0.06
experimentally.
n u m e r i c a l l y and D = 2 . 0 1 ± 0.10
Cluster aggregation
Some a s p e c t s o f r e s t r u c t u r i n g
by
deformation
o f the 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 p r o c e s s , 18 been s t u d i e d by Meakin and J u l l i e n discussed
If,
and are
is
irreversibi
i n the above m o d e l , we a l l o w
breaking
o f c l u s t e r s so t h a t a s t e a d y s t a t e e x i s t s example
not
here.
The l a s t p o i n t to i n v e s t i g a t e lity.
have
: each bond has
the f r a c t a l
a f i n i t e l i f e time) 19
d i m e n s i o n o f the c l u s t e r s
c l o s e to the f r a c t a l
(for
dimension of
is
very
The problem which a r i s e s i s to know i f two models ( r e v e r s i b l e
C l - C l and l a t t i c e
l e a d to the same u n i v e r s a l i t y moment, o n l y the f r a c t a l
class.
dimension i s
For
(randomly c h o s e n ) o f the
and so o n . S t a u f f e r
unoccupied
(Eden-type a g g r e g a t i o n ) ,
found the same f r a c t a l
R e v e r s i b l e DLA model has been s t u d i e d by two 22 o f us . I n t h i s v e r s i o n o f the p a r t i c l e - c l u s t e r 23 aggregation without
, we s t a r t
from any connected
l o o p s , on a l a t t i c e .
We take random
the
the c l u s t e r a g a i n ( D L A - t y p e a g g r e g a t i o n ) .
(percolation).
i s a model o f connected a g g r e g a t e s where we
b e g i n s a random walk u n t i l i t
To have a l o o p l e s s s t r u c t u r e ,
we d e c i d e
s t i c k i n g a r i s e s when the d i f f u s i v e
s a y B . Then p a r t i c l e
It
several
that
particle,
say
of
the
A b a c k s up to
its
l a s t brownian s t e p and we d e c i d e i t o n l y to p a r t i c l e
the
s t i c k s to
A , reaches a s i t e o c c u p i e d by a p a r t i c l e cluster,
clus
i n the s e t o f the s i n g l y connected
particle
other
di
mension as f o r l a t t i c e a n i m a l s .
animals)
a g g r e g a t i o n - f r a g m e n t a t i o n models a t
context
at a
o n e s . We break the c o r r e s p o n d i n g b o n d , then
the
equilibrium ? R e v e r s i b l e Eden model has been s t u d i e d by 21 i n another
t h i s does not d i s c o n n e c t
s u r f a c e o f the c l u s t e r
available
to t e s t t h i s c o r r e s p o n d a n c e . What about
Stauffer
point
(if
and then put t h i s p a r t i c l e
ly a particle
animals .
kinetic
remove a p a r t i c l e the c l u s t e r )
ter
lattice
2 0
257
Β o f the c l u s t e r
is
connected
(A can have
n e i g h b o u r s , b u t o n l y one bond (between
II
4-4 4-4 4-4 < >~4
160
36000
FIGURE 2 S k e t c h o f r e v e r s i b l e DLA p r o c e s s a c t i n g on a compact l o o p l e s s c l u s t e r (A) o r a t y p i c a l l o o p l e s s DLA c l u s t e r (B) o f 100 p a r t i c l e s . The number o f i t e r a t i o n s i s i n d i c a t e d below the f i g u r e s , and grows from l e f t to r i g h t .
R. Botet et al.
258
A and B) i s f o r m e d ) . T h i s i s a p r o c e s s a l r e a d y 25 i n t r o d u c e d by K a d a n o f f
f o r DLA .
S t a r t i n g from any i n i t i a l t o the same s t a t i s t i c a l f o r very l a r g e t i m e s . fractal
configuration
distribution
leads
of c l u s t e r s
In t h i s steady s t a t e ,
the
dimension i s found to be very c l o s e to
t h a t o f l a t t i c e a n i m a l s . The f o u r r e v e r s i b l e mo dels
(lattice animals, reversible
ble Eden, r e v e r s i b l e
DLA) are a l l
the sense t h a t the s t a t i s t i c a l finite cluster
CI-CI,
reversi
different,
i s d i f f e r e n t i n each model. T h i s
does not mean, however,
t h a t the s c a l i n g p r o p e r
t i e s f o r very l a r g e c l u s t e r are d i f f e r e n t . fractal
dimensions suggest t h a t a l l
s i b l e models b e l o n g to the same class cal
in
weight of a given
these
rever
universality
( l a t t i c e a n i m a l s ) . B u t so f a r
arguments have s u p p o r t e d t h i s
The
no t h e o r e t i
result.
We acknowledge the c o l l a b o r a t i o n and d i s c u s s i o n s w i t h H. Hermann and P . M e a k i n . T h i s work has been s u p p o r t e d by an ATP C . N. R. S . and by the CCVR, P a l a i s e a u . REFERENCES 1 . A review on a g g r e g a t i o n p r o c e s s can be found 1 i n " k i n e t i c s o f A g g r e g a t i o n and G e l a t i o n , e d s . F. Family and D. P. Landau ( N o r t h H o l l a n d 1984) 2. S . R. F o r r e s t and T. A . W i t t e n J r . , J . P h y s . A 12, L 109 (1979) 3 . D . " X Weitz and M. O l i v e r i a , P h y s . R e v . L e t t . 5 2 , 1433 (1984) TT7 A . W e i t z , Μ. Y . L i n and C . J . S a n d r o f f , S u r f a c e S c i . , 158, 147 (1985) 4 . P . M e a k i n , P h y s . Rev. L e t t . , 5 1 , 1119 (1983) 5 . M. K o l b , R. B o t e t and R. J u l l T e n , P h y s . R e v . L e t t . , 5 1 , 1123 (1983) 6. Recent experiments have been done i n a twod i m e n s i o n a l c o n f i n e d space ( l a y e r ) . They a r e d e s c r i b e d i n : A . Hurd and D. S c h a e f e r , P h y s . Rev. L e t t . 5 4 , 1043 ( 1 9 8 5 ) . A p o s s i b l e e x p l a n a t i o n o f t f i e i r r e s u l t s has been g i v e n i n : R. Jul l i e n , to be p u b l i s h e d 7. R. J u l l i e n , M. K o l b and R. B o t e t , J . P h y s i q u e L e t t . 4 5 , L 211 (1984) P . MeaTTn, P h y s . L e t t . A J O 7 A , 269 (1985) 8 . R. C. B a l l and T. A . W i t t e n , P h y s . Rev. A 2 9 , 2966 (1983) R. C . B a l l , J . S t a t . P h y s . 3 6 , 873 (1984) S . P. Obukhov, ' K i n e t i c a l l y a g g r e g a t e d c l u s t e r s ' , p r e p r i n t (1984) R. B o t e t , J . P h y s . A 1 8 , 847 (1985)
9 . P. M e a k i n , J . Chem. P h y s . j M , 4637 (1984) 10. R. B o t e t , R. J u l l i e n and M. K o l b , P h y s . Rev. A 3 0 , 2150 (1984) P. M e a k i n , J . C o l l o i d and I n t e r f a c e S c i . 102, 491 (1984) 1 1 . R. B o t e t and R. J u l l i e n , J . P h y s . A 17, 2517 (1984) M. K o l b , P h y s . Rev. L e t t . 5 3 , 1653 (1984) P . M e a k i n , T. V i c s e k and F T " F a m i l y , P h y s . Rev. Β 3 1 , 564 (1985) 12. J . P . U T e v a l i e r , C . C o l l i e x , M. Tence, R. J u l l i e n and R. B o t e t , ' F r a c t a l s t r u c t u r e o f p o l y d i s p e r s e i r o n a g g r e g a t e s : STEM a n a l y s i s and numerical s i m u l a t i o n s ' , i n p r e p a r a t i o n 13. J . P. C h e v a l i e r , C C o l l i e x and M. T e n c e , ' A n a l y s i s o f d i g i t a l i z e d STEM m i c r o g r a p h s : A p p l i c a t i o n t o the c a l c u l a t i o n o f the f r a c t a l dimension of iron a g g r e g a t e s ' , poster presen ted to : C o l l o q u e annuel de l a S o c i e t e F r a n g a i s e de m i c r o s c o p i e e l e c t r o n i q u e , S t r a n s b o u r g 28-31 mai 1985 14. M. K o l b and H. J . Herrmann, J . P h y s . A 1 8 , L 435 (1985) 15. P. M e a k i n , J . C o l l o i d and I n t e r f a c e S c i . 102, 505 (1984) P . M e a k i n , P h y s . R e v . A 2 9 , 997 (1984) R. C . B a l l and R. J u l l i e n , J . P h y s i q u e L e t t . 4 5 , L 1031 (1984) 1 6 Γ Ε . J u l l i e n and M. K o l b , J , P h y s . A 17, L 639 (1984) M. K o l b and R. J u l l i e n , J . P h y s i q u e L e t t . 45 L 977 (1984) 17. D. A . W e i t z , J . S . H u a n g , Μ. Y . L i n and J . S u n g , 'The l i m i t s o f the f r a c t a l dimension for i r r e v e r s i b l e k i n e t i c aggregation of c o l l o i d s ' , p r e p r i n t (1985) 18. P . M e a k i n and R. J u l l i e n , J . P h y s i q u e 4 6 , 1543 (1985) 19. M. K o l b , ' R e v e r s i b l e d i f f u s i o n l i m i t e d c l u s t e r a g g r e g a t i o n ' , p r e p r i n t (1985) 20. H . P . P e t e r s , D. S t a u f f e r , H. P . H b l t e r s and K. L o e w e n i c h , Z . P h y s i k Β 3 4 , 399 (1979) Β . D e r r i d a and L. de S e z e , T . P h y s i q u e 4 3 , 475 (1982) V. P r i v m a n , F. F a m i l y and A . M a r g o l i n a , J . P h y s . A 1 7 , 2837 (1984) 2 1 . D. S t a u T f e r , P h y s . Rev. L e t t . 4 1 , 1333 (1978) 22. R. B o t e t and R. J u l l i e n , ' D i f f u s i o n l i m i t e d aggregation with d i s a g g r e g a t i o n ' , preprint (1985) 2 3 . For a r e v i e w , see r e f . 1 and p r o c e e d i n g s o f G e i l o ( ' S c a l i n g phenomena i n d i s o r d e r e d s y s t e m s ' , e d . R. Pynn ( 1 9 8 5 ) ) , Les Houches ( " F i n e l y d i v i d e d m a t t e r ' , e d . M. Daoud (1985)), and C a r g e s e ( ' O n growth and f o r m s ' e d s . Η. E . S t a n l e y and N. O s t r o w s k y , M a r t i n u s N i g h o f f publishers (1985)). 24. Note t h a t DLA c l u s t e r s —o f f - l a t t i c e have no l o o p s . S o l o o p s a r e an a r t e f a c t o f the lattice. 25.
Leo P. K a d a n o f f , J . S t a t . (1985)
Phys. 39,
267
255
CLUSTER AGGREGATION
R. BOTET, R. J U L L I E N and M. KOLB Laboratoire France
de P h y s i q u e de$. S o l i d e s , Ba*t. 510,
Universite
P a r i s - S u d , C e n t r e d ' O r s a y , 91405 O r s a y ,
We i n t r o d u c e the c l u s t e r i n g o f c l u s t e r s p r o c e s s a s a model to d e s c r i b e a g g r e g a t i o n o f c o l l o i d a l o r a e r o s o l p a r t i c l e s i n the low c o n c e n t r a t i o n r e g i m e . We i n v e s t i g a t e m o d i f i c a t i o n s o f the parameters of the model. I n p a r t i c u l a r , v a r i o u s r e v e r s i b l e models o f a g g r e g a t i o n are d e t a i l e d and we d i s c u s s p o s s i b l e r e l a t i o n s between them ( u n i v e r s a l i t y c l a s s e s ) . 1 . INTRODUCTION
an i n f i n i t e medium, s t a t i s t i c a l l y homogeneous,
The f o r m a t i o n o f a g g r e g a t e s by c l u s t e r i n g isolated particles
p l a y s an i m p o r t a n t
numerous s c i e n t i f i c sols,
1
areas .
well-defined
in
been found u s i n g numerical
brownian
of monodisperse
particles.
they are on two n e i g h b o u r i n g s i t e s o f the .
e x p l a n a t i o n has
they s t i c k i r r e v e r s i b l y dimer.
s i m u l a t i o n s . We do
know why t h e s e a g g r e g a t e s are
d e n s i t y N Q/ L
When two o f t h e s e p a r t i c l e s c o l l i d e ( i . e . when
geometrical 2 3
on many l e n g t h - s c a l e s '
A beginning for a theoretical
not y e t
w i t h an i n i t i a l
In c o l l o i d s a n d aero
a g g r e g a t e s show a s e l f - s i m i l a r
structure,
role
3
of
which i s a
function
o f i t s m a s s . When two c l u s t e r s c o l l i d e ,
but we b e g i n to know how they are f r a c t a l
and
s t i c k and form a l a r g e r r i g i d c l u s t e r ,
how t h e i r f r a c t a l
physics.
on. Since s t i c k i n g i s i r r e v e r s i b l e ,
d i m e n s i o n i s r e l a t e d to
I n the f o l l o w i n g , we i n t r o d u c e irreversible
cluster-cluster
which g i v e s a r e a l i s t i c
the model
of
aggregation ( C I - C I )
d e s c r i p t i o n of
colloid
rigid
T h i s c l u s t e r a l s o f o l l o w s a random walk
trajectory with a v e l o c i t y
fractal
lattice)
and form a s m a l l
they and so
the
number
o f c l u s t e r s d e c r e a s e s w i t h time and the p r o c e s s ends when t h e r e remains o n l y one s i n g l e c l u s t e r i n the box.
and a e r o s o l a g g r e g a t i o n .
I n t h i s v e r s i o n , the u n d e r l y i n g l a t t i c e
for
2. THE MODEL OF CLUSTERING OF CLUSTERS
b i d s c o n t i n u o u s r o t a t i o n o f the c l u s t e r s a l o n g
T h i s model has been proposed i n d e p e n d e n t l y by 4 5 Meakin and by us . We d e s c r i b e here o n l y the
their trajectory.
3-dimensional
s i m u l a t i o n s have been done w i t h a
v e r s i o n o f t h i s model s i n c e i t
the most u s e f u l riments
for direct
. Nevertheless all
6 have been n u m e r i c a l l y
is
comparison w i t h expedimensions l e s s
than
s t u d i e d ^ as well as d i
mensions l a r g e r than the upper c r i t i c a l sion Log as f o l l o w s
I
The model
: i n a box L χ L χ L w i t h
is
have not been a l l o w e d to r o t a t e .
defined
tion
generalization
L i k e w i s e , the r e l a t i o n
between v e l o c i t y
mass i s unimportant
f o r the g e o m e t r i c a l
if
that .is i f
it
is realistic,
periodic
the v e l o c i t y
not an i n c r e a s i n g f u n c t i o n o f the m a s s ^ .
i n f i n i t e c u b i c l a t t i c e , we put randomly N Q i d e n
a f u n c t i o n o f time depends
Each o f t h e s e p a r t i c l e s
a random walk on the l a t t i c e . The model
follows simulates
F i g u r e 1 shows a t y p i c a l
is
Of
o f the c l u s t e r s as
on t h i s
relation
c l u s t e r o f 1024
t i c l e s , grown by t h i s p r o c e s s i n a box o f 70 χ 70 χ 70.
and
features, 1
boundary c o n d i t i o n s , which i s a p o r t i o n o f an
particles.
Off-lattice
i s reasonable .
c o u r s e , the s i z e d i s t r i b u t i o n
tical
clusters
o f the random w a l k , and they show t h a t c l u s t e r s have the same f r a c t a l s t r u c t u r e i f t h i s r o t a g
dimen-
Log36
o
I n the s i m u l a t i o n the
1 1
.
par
size
R. Botet et al.
256
When a v e r a g i n g o v e r a l a r g e number o f
clus
t e r s , Log-Log p l o t of radius of g y r a t i o n versus mass shows a s c a l i n g r e l a t i o n
o f the form :
radii.
length scales greater D i s the f r a c t a l
i s D = 1.78
than a few monomer
d i m e n s i o n and i t s
value
± 0.05.
the experimental
MODIFYING THE STARTING CONDITIONS I f we a l l o w p o l y d i s p e r s i t y o f the monomers
( b a l l s with a d i s t r i b u t i o n of r a d i i )
the same
k i n d o f s i m u l a t i o n s show t h a t the r e s u l t i n g a g g r e
Ν ^ RD for all
4.
T h i s v a l u e i s v e r y c l o s e to ? v a l u e s o f F o r r e s t and W i t t e n
g a t e s are s t i l l
fractal
o b j e c t s , w i t h the same 12
f r a c t a l d i m e n s i o n a s i n the monodisperse case These numerical r e s u l t s have been s u c c e s s f u l l y
.
compared w i t h r e c e n t experiments on p o l y d i s p e r s e 13 Fe a g g r e g a t e s . M o r e o v e r , as time goes o n , the effective density
:
I - 1 ^ R^"u i=all clusters L becomes o f o r d e r u n i t y , s i n c e the mean c l u s t e r r a d i u s R i n c r e a s e s w i t h t i m e . Then the screening i s i n e f f i c i e n t
because o f
We are i n the s o - c a l l e d k i n e t i c and the f r a c t a l me i s
brownian
entanglement.
g e l a t i o n regime
dimension t y p i c a l
of t h i s
regi
D = 1.75 ± 0 . 0 7 14 ( i n d = 2 ) .
5 . MODIFYING THE DIFFUSION CONDITIONS We can i m a g i n e a m o l e c u l a r ( i n s t e a d o f brow n i a n ) d i f f u s i o n o f the c l u s t e r s , where the mean FIGURE 1 T y p i c a l 3 - d i m e n s i o n a l c l u s t e r o f 1024 grown by C l - C l p r o c e s s on a l a t t i c e .
f r e e path i s o n l y l i m i t e d particles,
clusters.
quenched Fe v a p o r s , and o f
Weitz and O l i v e r i a 3 : D = 1.75
± 0.05
on g o l d
colloids.
The f r a c t a l
clusters
Here,
is interesting
t o note how few
parameters
are needed to r e c o v e r the experimental The s t a r t i n g c o n d i t i o n s a r e
results,
: monodisperse
p a r t i c l e s a t low c o n c e n t r a t i o n . The d i f f u s i o n c o n d i t i o n s are
: brownian
diffusion. The s t i c k i n g c o n d i t i o n s a r e
: o n c e two
t e r s c o l l i d e , they s t i c k i r r e v e r s i b l y new r i g i d
other.
more deeply i n t o each
d i m e n s i o n o f the
i s D = 1.91 ±
resulting
0 . 0 3 1 .5
6 . MODIFYING THE S T I C K I N G CONDITIONS
3 . PHYSICAL PARAMETERS OF THE MODEL It
effi
c i e n t than i n the brownian case and on average the c l u s t e r s p e n e t r a t e
D=1.8±0.1on
by c o l l i s i o n s between
Here a l s o the s c r e e n i n g i s l e s s
clus
and form a
cluster.
What happens when we change some o f the meters o f the model ?
t h r e e parameters can be m o d i f i e d :
p r o b a b i l i t y o f s t i c k i n g when c o l l i d i n g , the d i t y o f the c l u s t e r s and the i r r e v e r s i b i l i t y
of
the s t i c k i n g p r o c e s s . I f we l e t
the s t i c k i n g p r o b a b i l i t y
tend to
we o b t a i n the chemical model. I n t h i s c a s e ,
0,
two
c l u s t e r s must c o l l i d e many t i m e s b e f o r e s t i c k i n g . B u t once s t u c k , i r r e v e r s i b i l i t y
implies
that
c l u s t e r s can not b r e a k . T h i s chemical model has
1 been s t u d i e d n u m e r i c a l l y
para
the rigi
(the
fi and
17
experimentally
h e i g h t o f the r e p u l s i v e b a r r i e r
between
two g o l d c o l l o i d s can be v a r i e d c h e m i c a l l y ) . The fractal D = 2.00
d i m e n s i o n o f the r e s u l t i n g c l u s t e r s i s ± 0.06
experimentally.
n u m e r i c a l l y and D = 2 . 0 1 ± 0.10
Cluster aggregation
Some a s p e c t s o f r e s t r u c t u r i n g
by
deformation
o f the 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 p r o c e s s , 18 been s t u d i e d by Meakin and J u l l i e n discussed
If,
and are
is
irreversibi
i n the above m o d e l , we a l l o w
breaking
o f c l u s t e r s so t h a t a s t e a d y s t a t e e x i s t s example
not
here.
The l a s t p o i n t to i n v e s t i g a t e lity.
have
: each bond has
the f r a c t a l
a f i n i t e l i f e time) 19
d i m e n s i o n o f the c l u s t e r s
c l o s e to the f r a c t a l
(for
dimension of
is
very
The problem which a r i s e s i s to know i f two models ( r e v e r s i b l e
C l - C l and l a t t i c e
l e a d to the same u n i v e r s a l i t y moment, o n l y the f r a c t a l
class.
dimension i s
For
(randomly c h o s e n ) o f the
and so o n . S t a u f f e r
unoccupied
(Eden-type a g g r e g a t i o n ) ,
found the same f r a c t a l
R e v e r s i b l e DLA model has been s t u d i e d by two 22 o f us . I n t h i s v e r s i o n o f the p a r t i c l e - c l u s t e r 23 aggregation without
, we s t a r t
from any connected
l o o p s , on a l a t t i c e .
We take random
the
the c l u s t e r a g a i n ( D L A - t y p e a g g r e g a t i o n ) .
(percolation).
i s a model o f connected a g g r e g a t e s where we
b e g i n s a random walk u n t i l i t
To have a l o o p l e s s s t r u c t u r e ,
we d e c i d e
s t i c k i n g a r i s e s when the d i f f u s i v e
s a y B . Then p a r t i c l e
It
several
that
particle,
say
of
the
A b a c k s up to
its
l a s t brownian s t e p and we d e c i d e i t o n l y to p a r t i c l e
the
s t i c k s to
A , reaches a s i t e o c c u p i e d by a p a r t i c l e cluster,
clus
i n the s e t o f the s i n g l y connected
particle
other
di
mension as f o r l a t t i c e a n i m a l s .
animals)
a g g r e g a t i o n - f r a g m e n t a t i o n models a t
context
at a
o n e s . We break the c o r r e s p o n d i n g b o n d , then
the
equilibrium ? R e v e r s i b l e Eden model has been s t u d i e d by 21 i n another
t h i s does not d i s c o n n e c t
s u r f a c e o f the c l u s t e r
available
to t e s t t h i s c o r r e s p o n d a n c e . What about
Stauffer
point
(if
and then put t h i s p a r t i c l e
ly a particle
animals .
kinetic
remove a p a r t i c l e the c l u s t e r )
ter
lattice
2 0
257
Β o f the c l u s t e r
is
connected
(A can have
n e i g h b o u r s , b u t o n l y one bond (between
II
4-4 4-4 4-4 < >~4
160
36000
FIGURE 2 S k e t c h o f r e v e r s i b l e DLA p r o c e s s a c t i n g on a compact l o o p l e s s c l u s t e r (A) o r a t y p i c a l l o o p l e s s DLA c l u s t e r (B) o f 100 p a r t i c l e s . The number o f i t e r a t i o n s i s i n d i c a t e d below the f i g u r e s , and grows from l e f t to r i g h t .
R. Botet et al.
258
A and B) i s f o r m e d ) . T h i s i s a p r o c e s s a l r e a d y 25 i n t r o d u c e d by K a d a n o f f
f o r DLA .
S t a r t i n g from any i n i t i a l t o the same s t a t i s t i c a l f o r very l a r g e t i m e s . fractal
configuration
distribution
leads
of c l u s t e r s
In t h i s steady s t a t e ,
the
dimension i s found to be very c l o s e to
t h a t o f l a t t i c e a n i m a l s . The f o u r r e v e r s i b l e mo dels
(lattice animals, reversible
ble Eden, r e v e r s i b l e
DLA) are a l l
the sense t h a t the s t a t i s t i c a l finite cluster
CI-CI,
reversi
different,
i s d i f f e r e n t i n each model. T h i s
does not mean, however,
t h a t the s c a l i n g p r o p e r
t i e s f o r very l a r g e c l u s t e r are d i f f e r e n t . fractal
dimensions suggest t h a t a l l
s i b l e models b e l o n g to the same class cal
in
weight of a given
these
rever
universality
( l a t t i c e a n i m a l s ) . B u t so f a r
arguments have s u p p o r t e d t h i s
The
no t h e o r e t i
result.
We acknowledge the c o l l a b o r a t i o n and d i s c u s s i o n s w i t h H. Hermann and P . M e a k i n . T h i s work has been s u p p o r t e d by an ATP C . N. R. S . and by the CCVR, P a l a i s e a u . REFERENCES 1 . A review on a g g r e g a t i o n p r o c e s s can be found 1 i n " k i n e t i c s o f A g g r e g a t i o n and G e l a t i o n , e d s . F. Family and D. P. Landau ( N o r t h H o l l a n d 1984) 2. S . R. F o r r e s t and T. A . W i t t e n J r . , J . P h y s . A 12, L 109 (1979) 3 . D . " X Weitz and M. O l i v e r i a , P h y s . R e v . L e t t . 5 2 , 1433 (1984) TT7 A . W e i t z , Μ. Y . L i n and C . J . S a n d r o f f , S u r f a c e S c i . , 158, 147 (1985) 4 . P . M e a k i n , P h y s . Rev. L e t t . , 5 1 , 1119 (1983) 5 . M. K o l b , R. B o t e t and R. J u l l T e n , P h y s . R e v . L e t t . , 5 1 , 1123 (1983) 6. Recent experiments have been done i n a twod i m e n s i o n a l c o n f i n e d space ( l a y e r ) . They a r e d e s c r i b e d i n : A . Hurd and D. S c h a e f e r , P h y s . Rev. L e t t . 5 4 , 1043 ( 1 9 8 5 ) . A p o s s i b l e e x p l a n a t i o n o f t f i e i r r e s u l t s has been g i v e n i n : R. Jul l i e n , to be p u b l i s h e d 7. R. J u l l i e n , M. K o l b and R. B o t e t , J . P h y s i q u e L e t t . 4 5 , L 211 (1984) P . MeaTTn, P h y s . L e t t . A J O 7 A , 269 (1985) 8 . R. C. B a l l and T. A . W i t t e n , P h y s . Rev. A 2 9 , 2966 (1983) R. C . B a l l , J . S t a t . P h y s . 3 6 , 873 (1984) S . P. Obukhov, ' K i n e t i c a l l y a g g r e g a t e d c l u s t e r s ' , p r e p r i n t (1984) R. B o t e t , J . P h y s . A 1 8 , 847 (1985)
9 . P. M e a k i n , J . Chem. P h y s . j M , 4637 (1984) 10. R. B o t e t , R. J u l l i e n and M. K o l b , P h y s . Rev. A 3 0 , 2150 (1984) P. M e a k i n , J . C o l l o i d and I n t e r f a c e S c i . 102, 491 (1984) 1 1 . R. B o t e t and R. J u l l i e n , J . P h y s . A 17, 2517 (1984) M. K o l b , P h y s . Rev. L e t t . 5 3 , 1653 (1984) P . M e a k i n , T. V i c s e k and F T " F a m i l y , P h y s . Rev. Β 3 1 , 564 (1985) 12. J . P . U T e v a l i e r , C . C o l l i e x , M. Tence, R. J u l l i e n and R. B o t e t , ' F r a c t a l s t r u c t u r e o f p o l y d i s p e r s e i r o n a g g r e g a t e s : STEM a n a l y s i s and numerical s i m u l a t i o n s ' , i n p r e p a r a t i o n 13. J . P. C h e v a l i e r , C C o l l i e x and M. T e n c e , ' A n a l y s i s o f d i g i t a l i z e d STEM m i c r o g r a p h s : A p p l i c a t i o n t o the c a l c u l a t i o n o f the f r a c t a l dimension of iron a g g r e g a t e s ' , poster presen ted to : C o l l o q u e annuel de l a S o c i e t e F r a n g a i s e de m i c r o s c o p i e e l e c t r o n i q u e , S t r a n s b o u r g 28-31 mai 1985 14. M. K o l b and H. J . Herrmann, J . P h y s . A 1 8 , L 435 (1985) 15. P. M e a k i n , J . C o l l o i d and I n t e r f a c e S c i . 102, 505 (1984) P . M e a k i n , P h y s . R e v . A 2 9 , 997 (1984) R. C . B a l l and R. J u l l i e n , J . P h y s i q u e L e t t . 4 5 , L 1031 (1984) 1 6 Γ Ε . J u l l i e n and M. K o l b , J , P h y s . A 17, L 639 (1984) M. K o l b and R. J u l l i e n , J . P h y s i q u e L e t t . 45 L 977 (1984) 17. D. A . W e i t z , J . S . H u a n g , Μ. Y . L i n and J . S u n g , 'The l i m i t s o f the f r a c t a l dimension for i r r e v e r s i b l e k i n e t i c aggregation of c o l l o i d s ' , p r e p r i n t (1985) 18. P . M e a k i n and R. J u l l i e n , J . P h y s i q u e 4 6 , 1543 (1985) 19. M. K o l b , ' R e v e r s i b l e d i f f u s i o n l i m i t e d c l u s t e r a g g r e g a t i o n ' , p r e p r i n t (1985) 20. H . P . P e t e r s , D. S t a u f f e r , H. P . H b l t e r s and K. L o e w e n i c h , Z . P h y s i k Β 3 4 , 399 (1979) Β . D e r r i d a and L. de S e z e , T . P h y s i q u e 4 3 , 475 (1982) V. P r i v m a n , F. F a m i l y and A . M a r g o l i n a , J . P h y s . A 1 7 , 2837 (1984) 2 1 . D. S t a u T f e r , P h y s . Rev. L e t t . 4 1 , 1333 (1978) 22. R. B o t e t and R. J u l l i e n , ' D i f f u s i o n l i m i t e d aggregation with d i s a g g r e g a t i o n ' , preprint (1985) 2 3 . For a r e v i e w , see r e f . 1 and p r o c e e d i n g s o f G e i l o ( ' S c a l i n g phenomena i n d i s o r d e r e d s y s t e m s ' , e d . R. Pynn ( 1 9 8 5 ) ) , Les Houches ( " F i n e l y d i v i d e d m a t t e r ' , e d . M. Daoud (1985)), and C a r g e s e ( ' O n growth and f o r m s ' e d s . Η. E . S t a n l e y and N. O s t r o w s k y , M a r t i n u s N i g h o f f publishers (1985)). 24. Note t h a t DLA c l u s t e r s —o f f - l a t t i c e have no l o o p s . S o l o o p s a r e an a r t e f a c t o f the lattice. 25.
Leo P. K a d a n o f f , J . S t a t . (1985)
Phys. 39,
267