Growth Dynamics of Rotating DLAClusters - PowerPoint PPT Presentation

Title: Growth Dynamics of Rotating DLAClusters


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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
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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, Jing Zhao et al., 2003.

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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
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Depending on embedding dimension De the fractal
dimension Df takes the following
values We propose a statistical annular
model of the fractal growth for De 2 and De 3
Euclidean spaces (De gt 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.
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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 rnnd, n 0,1,...,N,
and d is the width of rings.
Thus, a total number of particles in the n-th
layer (ring) is
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
be a jump probability to the (n1)-th layer
be a jump probability to the (n-1)-th
layer. Obviously,
is a probability for the particle to be in n-th
layer. In general,
Now we can obtain the adsorption probability for
a particle in the n-th layer
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If the particle is not adsorbed, then with a
probability q 1 Pn it jumps into an another
place. Then
where q-1, q0 and q1 are the probabilities of
jumps without adsorption into the corresponding
layer. Obviously, if N M then q 0.
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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

• goes to (n1)-th layer with probability
• goes to (n-1)-th layer with probability
• stays inside n-th layer with probability

Thus
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
Probabilities of the particle jumps under the
condition that there is no adsorption
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.
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Using the known problem of random sequential
adsorption (RSA) we find the probability of
adsorption of a particle after the jump to
another layer
where ? is a step-function, and i1 is the
particle jumps to the (n1)-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)
If rn rn1 rn-1 then
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4. Results
Df as a function of the cluster size
w 0.01
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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.
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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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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 DLA-cluster in more detail the
growth-site probability density should be
calculated. This is what we intend to do in the
nearest future.