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Growth Dynamics of Rotating DLA-ClustersGrowth Dynamics of

Rotating DLA-Clusters

Contents:

1. Introduction2. A radial-annular model3. Rotating clusters4. Results5. Computer simulation of rotating clusters6. Concluding remarks

A.Loskutov, A.Ryabov,D.Andrievsky, V.IvanovMoscow State University

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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", RecreationCenter 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 SocialSciences". 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:

1

1000

, Tg = 1.70

w=0

M

Rw

100

10

1

10 100

5

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 > 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 velocityis found.

De Df

2 1.71 3 2.49 4 3.40 5 4.33

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r n

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.

Thus, a total number of particlesin the n-th layer (ring) is

where [] is an integer part. Let Nn be a real number of particles in a n-th layer.

,2 rnMn d

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To formalize the pattern formation let

11 13

dp rn n

11 13

dp rn n

103

pn

.1 1 , 1, 1,0, 13

kdkp k p kr nn n k

be a jump probability to the (n+1)-th layer;

be a jump probability to the (n-1)-th layer.Obviously,

is a probability for the particle to be inn-th layer. In general,

Now we can obtain the adsorption probability for a particle in the n-th layer:

.1 0 11 1

1 1

N NNn n nP p p pn n n nM M Mnn n

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If the particle is not adsorbed, then with a probability q = 1 – Pn it jumps into an another place. Then

,1 1 0 111

1 1

Ninq p q q qnMi n

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.

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

1 ,nQ

1 ,nQ

0 .nQ

Thus1

11

0

1

11

1 11

,3 1

11

,3 1

1 11

.3 1

n

n nn

n

n

nn

n

n

n nn

n

N dM r

QP

NM

QP

N dM r

QP

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 12 2

1 1

1 1 11 1 .

3 3 3n n n

nn n n n n

N N Nd dP

M r M M r

Probabilities of the particle jumps under the condition that there is no adsorption:

12

11

0

12

11

1 11

,3 1

11

,3 1

1 11

.3 1

n

n nn

n

n

nn

n

n

n nn

n

N dM r

QP

NM

QP

N dM r

QP

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:

1

1

2

1

2 1( ) 1

2 2

2 1 ,

n i

n i

N

n n i niA N

n i n n i

n i n i n

r N d rp n

r r N d

r N d r

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 adsorptionin the n-th layer is:(remind: a rotating cluster is considered)

1

1

1( ) 1 .

3i

n Ai n i

iP p n

r

If rn ~ rn+1 ~ rn-1 then

1

2

2 1( ) 1

2 2

2 1 ,

( ).

n

n

N

n n nA N

n n n

n n n

n A

r N d rp n

r r N d

r N d r

P p n

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4. Results

Df as a function of the cluster size

= 0.01

10 100 1000 2

3

4

5

6

7

2D

3D

log

M(r)

r

The mass of the fractal as its function of the radius for 2D and 3D spaces.

200 400 600 800

1.8

7

6

5

4

3 2

1

D

r

1.7

1.6

1.5

1.4

1.3

1.2

1.1

1.0

0.9

Dependence of the fractal dimension on the fractal radius for different rotating frequencies.

5. Computer simulationThe 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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17

18

19

= 4.5

1

100

III

II

I

, Tg = 0.8 w=4.5

M

Rw

, Tg = 1.221

, Tg = 2

10

1

10 100

Starting from some rotation velocity ( 0.2) there isa 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 emptyspace inside the window starts to play an important role.

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0

2,0

I region

II region

III region

103

Df

1,8

1,6

1,4

1,2

1,0

0,8

1 2 3 4 5

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 DLA-cluster in more detail the growth-site probability density should be calculated. This is what we intend to do in the nearest future.