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VOLUME 31, NUMBER 4
PHYSICAL REVIEW A
APRIL 1985
Scaling in a ballistic aggregation
Shoudan Liang and Leo
model
P. Kadanoff
The James Franck Institute and Enrico Fermi Institute, The University
Chicago, illinois 60637
(Received 13 November 1984)
of Chicago, 5640 South Ellis
Avenue,
An aggregate is formed by a kinetic process in which particles move in parallel straight lines and
stick to a structure at the point of first contact. The corresponding structure looks like a fan. Two
different scaling regions are investigated: one in the body of the fan and the other in the fuzzy region near its edges.
Recently, considerable work' has been devoted to a kinetic process called diffusion-limited
or
aggregation,
DLA. To model the process, one places a random walker
in the neighborhood of an aggregate, then lets it walk until it hits. Then it sticks and the aggregate grows by one
unit. Simple as it is, the model is capa'ble of describing
some quite complicated spatial structures obtained by a
dynamic random growth process from an experiment. Because of its nonequilibrium
nature, DLA is beyond the
scope of application of conventional statistical mechanics.
For this reason, it still evades complete theoretical understanding.
DLA is only one among a large class of models of
nonequilibrium aggregation. What are the general properties of these models? With this question in mind, we studied another such model, what we call the ballistic driven
aggregation (BDA) model, due originally to Void and
In this model, particles are only allowed to
Sutherland.
move along straight lines un'til they hit the aggregate, and
stick. Randomness is introduced via a probabalistic
choice of the lines. Physically, this model may serve as an
approximation to deposition processes in the extreme high
vacuum limit in which particles move toward a substrate
in straight lines.
The model exhibits interesting behaviors of various
kinds. Previous workers ' have observed fractal effect
analogous to, but different from, that of DLA. We report, in this note, a scaling behavior not unlike what one
may see in a phase transition system.
In this paper, we concentrate upon the situation in
which all the lines are parallel so that the particles "rain"
in only from one direction (visualize this as downward
motion). In this case, we can simulate the system with a
very efficient computer algorithm. Since we only need to
keep track of the height of the top-most particle on each
line where particles move, computer memory space is not
a problem in constructing very large clusters. In fact, it is
possible to construct clusters of 10 million occupants with
moderate computer time.
The rules of the model are very simple: Imagine an aggregate lying between xi and x, . Choose, at random, a
value of x between xI+1 and x, —
1. A particle with this
value of x moves from
oo. It
oo toward y= —
sticks at the first site which has an occupied neighbor,
otherwise it disappears. Then another particle is shot in,
and so on. If one begins with an aggregate which contains
y=+
31
a single point, the outcome of this process is a fanlike
structure (Fig. I), which has a high density in the center
and a bunch of long holes near the edges.
To describe the "fan, one uses polar coordinates (r, 8),
in which the initial point lies at r =0 and 0=0 corresponds to the y direction. The lattice can be conveniently
divided into bins according to r and L9. In each bin, we
measure the average density defined as the ratio of the occupied sites to the available lattice sites.
Figure 2 shows the average density as a function of the
angle for several radii. The density varies smoothly at
small angle, and then it drops sharply to zero. The transition gets steeper when the radius approaches infinity.
Call the angle at which this transition occurs 0, . Then
one can divide the fan into three regions, depending upon
whether
8 &8„~8
8~ &8, . In the first re+ oo the density approaches a constant. What
gion, as r —
this means is that the fan structure has a trivial fractal dimension, which equals the dimensionality
of the space.
Nevertheless, the approach to the constant density is very
slow and takes a scale invariant form
"
~
~
~
=8„or
~
FICx. 1. An example of fanlike structure generated by BDA
model from a single seed. There are about 4)&10 particles in
this fan. The particles are rained from above. In the central
part, the particles are distributed almost uniformly. There are
not many of the big holes which are necessary for generating a
nontrivial fractal dimension.
Very long diagonal holes sometimes develop along the edge of the fan causing large density
fluctuation.
2628
1985
The American Physical Society
SCALING IN A BALLISTIC AGGREGATION MODEL
31
(1
p(r)
I
Ooo 00 0
Q~agq~&&
Qgg
000
QgiIg 0 0
4a
0.4—
op
4go
0 p
Q
ao
go
4
radius
o
radius
0
radius
p,
=p„+Ar
A, and P are constants which may depend on the angle. Here P is a nontrivial correction to fractal dimension.
The form (1) has been previously used by several other authors to analyze BDA in various different geometries.
To analyze the data, we averaged the density over sec8 —
tions (radius & r, 8 & angle & 8+). If
8+ is
small, this section density also obeys Eq. (1). We show in
Fig. 3 the density observed in our numerical simulations
for the particular sector in which 8+ —+10'. The data
can be fit with Eq. (1) over three decades. For this section
P=0.66+0.04. Gur simulation suggests but does not
7500
750
75
oo
o 0
0.3—
2629
~
e
o
0
0
0
0
n
0
0
no
40
0
0
0
~
0
0
0
0
0
0
0
0
0
04
0
0
00
op 0
I
0
40
30
20
10
I
P oonnn
50
r
o
o60
0
Q
0.32—
angle (degrees)
density as a function of angle at three different
Notice the transition to zero density becomes sharper
when the radius gets larger. At small angle, the density is
higher for small radii than large ones. The r =75 line is obtained by averaging over 2X10 clusters, and the line with
r =750 is the result of 2)&10 repetitions. The last line, radius
7500, is based on the data of 200 realizations of clusters of 10
million occupants. In each case, we extend the run sufficiently
to obtain a complete fan, so when more particles are added in,
the density at the radius used here will not be changed.
FIG. 2. The
radii.
+
+ 200
x 600
o 2000
4 6000
0
9b'
@+
0.24—
4
40
0
4
4
O. 1
X
0
X
X
0
4
x
0
X
0
0.08—
x
4
0
4
X
0
+
X
a
+
X
0
+
I
+++4-i
40
30
25
45
50
angle
0
0
0
r
0
x
0
0
o
+
p
o 60
+ 200
n
4. o
x+ pa
0
X
x
o
oa
0
0
Og
00
4 6000
4 x
0
X
0
n
0
0
+
x0
4
0
0
+o
+O
CA
C:
00
600
o 2000
0
+'k
QP
Xo
0
00X
0
0
0
0
0
0
0
0
SP
0
g3
~0
a
g
0'
FIG. 3. Sector
d (r) vs r. The plot shows
ln-ln scales.
The sector is
d is a parameter adjusted to
make the fit linear. The slope, which is the same as P in Eq. (1),
0. 66 while d =0.463. In this plot, the data for different
is —
regions are taken from the clusters of different sizes, with more
realizations for the smaller size clusters.
density
Ad = d (r) —
d „and r on
—10 &0&10'. The constant
I
-60
-100
-20
renarmalized
FIG. 4.'
20
60
100
angle
0„
for five values of r. (b)
(a) Density vs angle near
Scaling function in Eq. {2). The scaled density r"p(r, O) is plot8, ). The r values for
ted against the renormalized angle r (8 —
this plot are the same as that used in (a). Here, @=0. 1, v=0. 46,
and 0, =32.
0.
2630
SHOUDAN LIANG AND LEO P. KADANOFF
prove that as 8 increases P increases also.
When our work is taken in conjunction with the
analysis of Meakin, one can see quite strong evidence
that P is nonuniversal. Meakin considered a starting configuration in which all sites on the line y =0 were initially
occupied. Then he fit the density as a function of height,
h, to
~
~
p(r) =p„+Ar
Then, he found P=0. 78. When he changed the model by
allowing the particles to stick to next nearest neighbors he
found f3=0. 96. Clearly these are different answers from
our P value. The conclusion is that the value of P is
nonuniversal.
In the transition region near the edges of the fan, the
density goes quickly to zero, as 0 exceeds a critical value,
8, . Figure 4(a) suggests that 8, is about 32' and that near
8, there is a fairly large range over which p is linear in 8.
can be
The information in the region
compressed into a scaling function of the form
8~8„r~ao
p(r, 8)=r t'H(r"(8
8, )) .
—
(2)
in Fig. 4(b) which
We plot r"p(r, 8) versus r~(8 8, ) —
x ~.
shows that the function H(x)'is linear for small
This linearity allows us to determine the v —
p to be
0. 33+0.05. %'e determine the exponent p by using the
fact that at the critical point 8=8„p(r,8, ) is proportionThis gives 8, =32.0+0.5, and @=0.1+0.05.
al to r
Now we can see two different scaling behaviors, one for
8~ &8, and the other for 8=8, . The latter behavior is
31
probably dominated by very long thin holes near the
edges. These holes are quite apparent in Fig. 1. They are
produced by two regions which grow diagonally outward
almost in parallel, with the bottom region sticking out just
further than the top one. Between these two growing
layers, a hole develops. The correction to scaling and the
P value are harder to understand. Perhaps the correction
and its nonuniversality of P are due to the variation in the
angle between the normal to the surface and the y axis.
Call this angle P. We studied growth of surfaces held at
an average angle P away from the horizontal. We found
that this offset in angle tended to produce a surface with a
few very large steps in height which then moved across
the sample. The surface looks like a staircase with the
steps traveling from the high end to the lower one as the
cluster grows. This effect does lower the density because,
when the jumps travel, an empty region will be left
behind.
The big long shape holes near the edges of the fan (Fig.
1) not only cause the decrease in density but also increase
fiuctuations. Let Xb;„bethe number of occupied sites in
a bin. The data indicate that (Nb;„—
Nb;„) is proportional to Xbj„The proportionality constant is around 1 in
the center of the fan and is around the order of one hundred near the edges. This is why we could not determine
the angular dependence of P.
ACKNOWLEDGMENTS
~
".
~
T. A. Witten and L. M. Sander, Phys. Rev. Lett. 47, 1400
(1981); Phys. Rev. B 27, 5686 (1983).
P. Meakin, Phys. Rev. A 27, 604 (1983); 27, 1495 (1983); Phys.
Rev. Lett. 56, 1119 {1983).
M. T. Void, J. Colloid. Sci. 18, 684 (1963).
4D. N. Sutherland,
J. Colloid. Sci. 22, 300 (1966); 25, 373 (1967).
We .would like to thank D. Bensimon, C. Tang, B.
Shraiman, and P. Meakin for helpful discussions. This
research was supported by U. S. Department of Energy
Grant No. DESG0284ER45144 and made use of the Central Computational Facility of the University of Chicago,
Materials Research Laboratory.
5D. Bensimon,
238 (1984).
B. Shraiman,
and
S. Liang, Phys. Lett. 102A,
D. Bensimon, B. Shraiman, and L. P. Kadanoff,
of
D. P. Lan-
in Kinetics
Aggregation and Gelation, edited by F. Family and
dau (North-Holland, Amsterdam, 1984), p. 75.
7P. Meakin (private communication).