Transcript Заголовок слайда отсутствует

Growth Dynamics of
Rotating DLA-Clusters
A.Loskutov, A.Ryabov,
D.Andrievsky, V.Ivanov
Moscow State University
Contents:
1. Introduction
2. A radial-annular model
3. Rotating clusters
4. Results
5. Computer simulation of rotating clusters
6. Concluding remarks
1
Some publications
• A.Loskutov, D.Andrievsky, V.Ivanov, K.Vasiliev and
A.Ryabov. Fractal growth of rotating DLA-clusters.Macromol. Symp., 2000, v.160, p.239-248.
• A.Loskutov, D.Andrievsky, V.Ivanov, K.Vasiliev and
A.Ryabov. Annular model of rotational fractal clusters.- In
Proc. of the Int. Conf. "Nonlinear Dynamics in Polymer and
Related Fields", Recreation
Center DESNA, Moscow Region, Russia 11-15 October
1999. Ed. M.Tachia and A.R.Khokhlov, Japan, 2000, p.51-54.
• A.Loskutov, D.Andrievsky, V.Ivanov and .Ryabov.
Analysis of rotating DLA-clusters: Theory and computer
simulation.--- In: NATO ASI, Ser.A, vol.320 "Nonlinear
Dynamics in the Life and Social
Sciences". Eds. W.Sulis and I.Trofimova.- IOS Press 2001,
p.253--261.
• A.Loskutov, D.Andrievsky, V.Ivanov and A.Ryabov.
Growth dynamics of rotating DLA-clusters.- In: "Emergent
Nature". Proc. of the Int. Conf. "Fractal'2002", Granada,
Spain, March 2002. Ed. M.M.Novak.- World Scientific,
2002, p.263-272.
• A.Loskutov, D.Andrievsky, V.Ivanov and A.Ryabov.
Analysis of the DLA-process with gravitational interaction
of particles and growing cluster.- In: WAVELET
ANALYSIS AND ITS APPLICATIONS. Ed. Jian Ping Li,
2
Jing Zhao et al., 2003.
1. Introduction
Theoretical
and
computer
simulation
analysis of cluster growing by diffusion
limited aggregation under rotation around a
germ is presented. The theoretical model
allows to study statistical properties of
growing clusters in two different situations:
in the static case (the cluster is fixed), and in
the case when the growing structure has a
nonzero rotation velocity around its germ.
By
the
direct
computer
simulation
the
growth of rotating clusters is investigated.
The fractal dimension of such clusters as a
function of the rotation velocity is found. It
is shown that for small enough velocities the
fractal dimension is growing, but then, with
increasing rotation velocity, it tends to the
unit.
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The characteristic property
of fractal growth is that this
process is not described by
equations; the statistical models
are usually used. The most known
process of irreversible adsorption
is diffusion limited aggregation
(DLA). In the simplest case this is
a cluster formation when the
randomly moving particles are
sequentially
and
irreversibly
deposited on the cluster surface.
As a result of such clusterisation a
known DLA-fractal is formed:
w=0
1000
100
M
10
 , T g  = 1 .7 0
1
1
10
Rw
100
4
Depending on embedding
dimension De the fractal dimension
Df takes the following values:
De
Df
2
3
4
5
1 .7 1
2 .4 9
3 .4 0
4 .3 3
We propose a statistical
annular model of the fractal
growth for De = 2 and De = 3
Euclidean spaces (De > 3 to be
obtained). This model allows us
to find dimensions of fixed
(“classical DLA”) and rotating
DLA-clusters.
Main result: the fractal dimension Df
as a function of the angular velocity
is found.
5
2. A radial-annular model
Suppose that particles have a diameter d.
Let us divide an embedding space into N
concentric rings with a middle radius rn=nd,
n= 0,1,...,N, and d is the width of rings.
rn
Thus, a total number of particles
in the n-th layer (ring) is
2 rn 
Mn
,

d 




where [] is an integer part. Let Nn be a real
number of particles in a n-th layer.
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To formalize the pattern formation let

p1  1 1  d
n 3  rn




be a jump probability to the (n+1)-th layer;

p  1  1 1  d
n
3  rn




be a jump probability to the (n-1)-th layer.
Obviously,
0 1
p 
n 3
is a probability for the particle to be in
n-th layer. In general,


k
p k  1 1  k d  ,  p 1, k   1,0,  1 .
rn 
n
n 3
k
Now we can obtain the adsorption
probability for a particle in the n-th
layer:
N
N
N

1
0
Pn  n 1 p n  n p n  n 1 p1n
M
Mn
M
n 1
n 1
.
7
If the particle is not adsorbed, then with
a probability q = 1 – Pn it jumps into an
another place. Then


1  N n 1  i
1 q0  q1 ,
q   1 

q
 p
n


M
i  1
n 1 
where q-1, q0 and q+1 are the probabilities
of jumps without adsorption into the
corresponding layer.
Obviously, if N = M then q = 0.
8
Suppose that the particle is in a n-th
annular layer. It is adsorbed here with
probability Pn. If the adsorption in n-th
layer is not taken place then the particle
1
• goes to (n+1)-th layer with probability Q n ,
1
Q
• goes to (n-1)-th layer with probability n ,
• stays inside n-th layer with probability Q n0 .
Thus
1
Qn

N n 1
1


M n 1
1

3
1

d 
1



r
n 

,
Pn

Nn 
1 

M
1
n 

0
Qn 
,
3 1  Pn
1
Qn

N n 1
1 
M n 1
1

3
1

d 
 1  
rn 

.
Pn
The obtained expressions allow us to find
the cluster mass as a function of its radius.
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Similar ideas can be applied to 3D spaces.
In this case the adsorption probability is:
1 N n 1 
d  1 Nn
1 N n 1
Pn 

1  2  
3 M n 1 
rn  3 M n 3 M n  1

d 
1  2  .
rn 

Probabilities of the particle jumps under
the condition that there is no adsorption:
1
Qn

N n 1  
d 
1 
 1  2 
M n 1  
rn 
1

,
3
1  Pn

Nn 
1 

M
1
n 
0
Qn 
,
3 1  Pn
1
Qn

N n 1  
d 
1 
 1  2 
M n 1  
rn 
1

.
3
1  Pn
Here Mn is a total number of particles in
a spherical layer of radius rn.
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3. Rotating clusters
Let us find the adsorption probability.
The model: rotation is a continuous
process, but the particle makes jumps at a
time interval . Between jumps the particle
is at rest.
Let us pass to a rotational frame of
reference. Then during time  the particle is
moved as  = . In this case we have the
effective particle size: deff = d + . Suppose
that in the n-th layer there exist n particles
of a diameter d.
11
Using the known problem of random
sequential adsorption (RSA) we find the
probability of adsorption of a particle
after the jump to another layer:
p (n)  1 
i
A
 2 r
n 1
  N n  i  1  d    rn 
2 rn  i  2 rn  1  N n  i d 
  2 rn  i   N n  i  1  d    rn 
N n  i 1
N ni  2

,
where  is a step-function, and
i=1 is the particle jumps to the (n+1)-th layer,
i = -1 if the particle jumps to (n-1)-th layer,
i = 0 if the particle stays in n-th layer.
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The probability of the adsorption
in the n-th layer is:
(remind: a rotating cluster is considered)
Pn 
1
1

3
i  1

i 
p (n) 1 
.
rn  i 

i
A
If rn ~ rn+1 ~ rn-1 then
p A (n)  1 
 2 r   N
n
n
 1  d    rn 
2 rn  2 rn  N n d 
N n 1

Nn 2
  2 rn   N n  1  d    rn  ,
Pn  p A ( n ).
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4. Results
7
1.8
1
3
2
4
1.7
lo g M (r)
6
3D
1.6
1.5
2D
5
5
1.4
D 1.3
4
6
1.2
1.1
3
7
1.0
2
10
0.9
100
r
1000
The mass of the fractal as
its function of the radius
for 2D and 3D spaces.
200
400
r
600
800
Dependence of the fractal
dimension on the fractal
radius
for
different
rotating frequencies.
Df as a function of the cluster size
 = 0.01
14
5. Computer simulation
The method of sliding windows has been used for
calculating fractal dimension. 100 points have been
chosen randomly on the cluster. Each of them was
treated as the center of a square with the side Rw
from 1 to 100, and the number of occupied sites
inside that square was calculated. The dependence
M(Rw) was averaged over these 100 different
“windows” and over a quite enough number of
independent clusters.
16
17
18
III
w = 4 .5
 , T g  = 0 .8
II
100
M
10
I
 , T g  = 1 .2 2 1
, T g  = 2
1
1
10
Rw
100
 = 4.5
Starting from some rotation velocity (  0.2) there is
a clear bending point on the plot (regions I and II):
beginning with this velocity the width of the “tail”
becomes smaller than the maximal size of “sliding
windows” which is equal to 100. Therefore, by
measuring the number of occupied sites the empty
space inside the window starts to play an important role.
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2 ,0
1 ,8
I regio n
1 ,6
II regio n
1 ,4
III regio n
Df
1 ,2
1 ,0
0 ,8
0
1
2
3
1 0
4
5
3
Upon increasing  the slope of the line in the
region I goes to 2, and the slope in the region II
tends to the unit. The cluster becomes more
dense on small scale and degenerates into the
thin stripe or even into a line. At >1.5 one can
find even three regions (I, II and III). It turns
out that the slope in the region III is smaller
than the unit. This is, of course, an artefact and
it can be explained only as a finite size effect.
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6. Concluding remarks
Main result: we find the fractal dimension Df as a
function of the angular velocity for two-dimensional
DLA-cluster under rotation.
The fractal dimension decreases when the rotation
velocity is growing. We have found the clear evidence
of a transition from fractal (at low rotation velocities)
to non-fractal (at high velocities) regimes. The results
of our simulation partly correspond rather well to
those obtained by Lemke et al. (Phys.Rev.E, 1993) for
infinite initial mass case. We did not perform here a
detailed analysis of number of spiral arms. Instead,
we concentrated ourselves on studying the difference
in fractal behavior on different scales.
The DLA-cluster under rotation shows different
fractal dimensions when analyzed on different length
scales. To analyze the multifractal properties of DLAcluster in more detail the growth-site probability
density should be calculated. This is what we intend
to do in the nearest future.
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