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Anton Menshutin
, 2011, October


Two dimensional aggregate growth
Ice crystals

D3=2.2-2.6

D2=1.4-1.8


Two dimensional aggregate growth

D=1.5-1.8
Natural Cu

Goethite in agate

Dendrits & aggregates

Manganese oxide in chalcedony


Two dimensional aggregate growth

Bacteria colonia Bacillus subtilis from the site www.igmors.u-psud.fr

D=1.7


Two dimensional aggregate growth

D=1.6-1.9

Nano-crystals grown on a substrate


A model for two-dimensional aggregate growth -- DLA model
Diffusion limited aggregation ­ DLA Witten and Sander, PRL, 1981
1. Place seed at origin (0,0), N=1 2. Particle starts at radius of birth Rbirth 3. Diffusion in space 4. If collision occurs, it sticks. N =N + 1 5. If particles goes out of the radius of death Rdeath it is killed 6. New iteration ­ from step 2.

D=3/2 (?)
on the square lattice

D=1.72
Off-lattice


Problem definition

· Large class of models of 2D growth results in similar structures (weakly speaking - "looks similar") · DLA model is the simplified model that captures main features of 2D critical aggregate growth · Scaling exponents are known only with 1% accuracy · Cluster behavior as N is still not understood · Multiscaling?


Off-lattice killing-free algorithm
The most effective and most correct realization of algorithm Particles are balls of unit radius. A single particle at (0,0) position - a seed. At random point on birth circle Rb a new particle is born. Particle moves randomly. If particle goes out of Rd (Rd>Rd) it is returned to Rb at angle phi with probability

1 x 2- 1 W ( )= x= r / R 2 x 2- 2x cos +1

b

Repeat random walk until collision with cluster.


Off-lattice killing-free algorithm

50 000 000 particles 1000 clusters in each ensemble


Scaling in DLA D

NR

How to measure R? Take an average over cluster surface with weight dq (growth probability at point r) Numerically: freeze cluster, launch probe particles, record their position r_i and average it.

Rdep = r dq
R 22 =



r dq

2

Re =exp



ln ( r ) dq

1 R de p = M

M

1,712


1

r

R22
Limit

i

1,711 1,710 1,709
D

Rd Re

N
?????

1,708 1,707 1,706 0 1x10
7

2x10

7

3x10
N

7

4x10

7

5x10

7


Growth probability
Let P(r,N) be a probability for a next particle to be attached at distance r and N is a size of cluster.


Growth probability

Let us assume a scale invariant form of P(r,N)
1 0,1 0,01

1 P ( r , N )= f ( r / Rdep ) R de p

N=500000 N=16000000 N=50000000

P(x)

1E-3 1E-4 1E-5 1E-6 1E-7 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4

x=r/R

dep


Scaling assumption and consequences
1 P ( r , N )= f ( r / Rdep ) <=> R dep

Scaling indexes of Rdep, R22, Re ... are equal.

R dep= r P ( r , N ) dr R e =exp ln ( r ) P ( r , N ) dr
R 22 =

R



r P ( r , N ) dr

2

d ep

R 22 R

e


Multiscaling in DLA
Particle density at distance r from cluster origin

g ( r , N )= A ( x ) R

D ( x )- 2 gyr

, x= r / R

gyr

Non-trivial family of indexes D(y)?


Scaling assumption and consequences

No multiscaling
K=N

R

dep

N

1/D

g ( r , N )=

K =1



P ( r , K ) dK g ( r , N ) r


( D - 2)

F ( r / Rdep ( N ) ) dx

F ( t )= f ( x ) x
t

-D


Numerical check of multiscaling
ln g ( r , R gyr )= ln c ( y ) +( D ( y )-2 ) ln R
1,80
gyr

1,75

1,70

D (y )

1,65

1,60

N N N N

=[50;50mln] =[1000;50mln] =[1mln;50mln] =[10mln;50mln]

1,55 0,0 0,5 1,0 1,5 2,0

y


Radial density in scale invariant form
g ( r , N ) x
1000 100 10 1
( D- 2 )

F (r / R

dep

( N )) , x = r / R

dep

- +

x

1.7065-2

g ( r , N = 5 0 m ln )

0,1 0,01 1E-3 1E-4 1E-5 1E-6 1E-7 0,01 0,1 1

x=r/Rdep


Scaling function F(t) F ( t )= f ( x ) x-D dx
t

f(x)=P(r,N=50mln)
2-D

1,0

g(r,N)r integrated f(t)

0,8

F(t)

0,6

0,4

0,2

0,0 0,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

t


Conclusion
1) su 2) 3) 4) 5) 6) New scale-invariant form of growth probability P(r,N) ggested. "All" scaling exponents are equal to D. Multiscaling is finite size effect. ( D- 2 ) F ( r / R dep ( N )) , x = r / R Density of particles g ( r , N ) x No collapse as N DLA is a model of Self-organized criticality class.

dep

Suggestion: Need to derive P(r,N) on g(r,N) dependence for exact P(r,N) calculation.