EXPERIMENTAL AND SIMULATED
STUDY OF DIFFUSION LIMITED
AGGREGATION OF SUSPENDED
MAGNETIC MICROSPHERES
Group members:
Rabia Aslam Chaudary (12100011)
Aleena Tasneem Khan (12100127)
Supervisor:
Dr. Fakhar-ul-Inam
OUTLINE
• Diffusion Limited Aggregation
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What is DLA?
The DLA Model and it’s applications
Other models
Fractal Dimensions
• Our approach to the study:
– Experimental Study
– Simulated study
• Past studies done of DLA clusters
DIFFUSION LIMITED AGGREGATION
What is Diffusion Limited
Aggregation?
• Diffusion Limited Aggregation (DLA) is an algorithm of
simple growth in which a cluster grows when individual
particles are added to it through a diffusion-like process.
• Originally proposed by Witten and Sander in 1981, the
model is used to study wide variety of systems from
electrodeposited growth and dielectric breakdown to
formation of snow flakes and lightening paths.
USING THE DLA MODEL
a. Simulated DLA of about 33,000 particles.
b. High-voltage dielectric breakdown
c. Copper sulfate in an electro-deposition cell
• An animation of DLA, for the purpose of our project:
Chi-Hang Lam, Applied Physics, Hong Kong Polytechnic University
Fractal Dimensions
• Fractal dimension is a statistical quantity that
indicates how completely the fractal fills space.
• The geometrical pattern of fractals is repeated at
every small scale
• Fractals have non-integer dimension D.
log(no. of self similar pieces)
• Fractal Dimension =
log(magnification factor)
D 
ln  r 
ln( N )
Fractal Dimensions
• For clusters in a plane, (in 2D), the fractal dimension
D is bounded by the value D = 1.71
• For clusters in space, (in 3D), the fractal dimension D
is bounded by the value D = 2.5
• Fractal dimension is sensitive to the lattice structure
of the particle and to the environment of the
structure.
Other models:
• The Eden Growth model:
Growth of specific type of clusters
like bacterial colonies and deposition
of metals. Clusters growth by random
accumulation of material on their
boundary.
• The Ballistic Aggregation Model:
If the random walks of the particles are placed
by ballistic trajectories, we have the ballistic
Aggregation model. It generates non-fractal
Clusters characterized by a power law.
RECENT STUDIES OF THE DLA
CLUSTERS
• Diffusion-Limited Aggregation, a Kinetic Critical
Phenomenon (1981)
(T. A. Witten, Jr. and I. M. Sander)
Witten and Sander proposed the DLA model studying aggregates
formed when a metal vapor produced by heating a plated
filament was quench condensed.
• Model for the growth of electrodeposited
ferromagnetic aggregates under an in-plane magnetic
field (2010)
(C. Cronemberger, L. C. Sampaio, A. P. Guimarães, and P. Molho)
Effect of Increasing magnetic moment and external field on the
aggregates and fractal dimensions of ferromagnetic particles.
Aggregates by simulations at different values of magnetic moment and applied magnetic field
• Aggregation of Magnetic Microspheres: Experiments
and Simulations (1988)
(G. Helgesen, ' A. T. Skjeltorp, P. M. Mors, ' R. Botet, and R. Jullien)
Diffusion Limited cluster aggregation of magnetic microspheres.
Complete agreement of experiment and simulation.
Aggregates formed as a result of experiment as magnetic field increases from a to d.
Simulated Results
a. Without dipolar interactions
and rotational diffusion
b. Without dipolar interactions
but with rotational diffusion
c. With dipolar interactions
and rotational diffusion
d. Adding external magnetic
field
Our model for non-magnetic and
magnetic microspheres
• We are basing our model on original DLA model for both
types of particles.
• First particle is placed in the center. Other particles enter from
boundary of the cell undergoing a periodic boundary
condition and doing Brownian movement and sticks to make
aggregate.
• At each step, particles have four possibilities for its next
position and they are assigned probabilities accordingly.
• For magnetic particles, the dipole moment is given by:
 M(
d
3
)
6
• Magnetic interactions between two spheres, i and j, separated
by the distance rij  ri  r j ,is given by the following relation,
 u i .u j  3 ( u i .rij )( u j .rij ) 
D ij  3 

2
rij 
rij

• We also have two dimensionless parameters, effective strength
of dipole-dipole interactions and dipole-field interactions.

2
K dd 
K df 

2
3
d k BT
 B ext
k BT
• The total energy of a particle at the position ri is given by:
U mag

  
( ri )    i . B T ( ri )
• Differently from DLA, the energy difference between the
current position and the four possible new positions is used to
calculate the probabilities.
1
exp( 
Pi 
k BT
U i )
P
i
i
• According to this model, the particle moves to the region of
lower energy with higher probabilities.
EXPERIMENTAL SETUP FOR THE DLA
CLUSTER STUDY
Experimental Procedure
o Study of non-magnetic particles:
Particles doing Brownian motion observed by
microscope and camera. Possibility of cluster
aggregation.
o Study of magnetic particles:
Sulfonated polystyrene magnetic microspheres with 30%
iron oxide dispersed in water confined to a mono-layer.
Experimental setup:
• Setup to vary temperature
• Application of External Field
CONTROL PARAMETERS
•
•
•
•
•
Seed Size
Doping
Solvent
External Magnetic field
Temperature
To study:
The effect on Fractal dimensions and scaling properties of the
aggregated clusters
SIMULATED STUDY OF THE DLA
CLUSTER MODEL
Outline of simulation
FORMATION OF LATTICE
AND INTRODUCTION OF
SEED
LOOP OVER THE DESIRED
NUMBER OF PARTICLES
UNTIL A CLUSTER IS
FORMED
CALCULATE FRACTAL DIMENSION
BY CALCULATING THE RATIO OF
NUMBER OF PARTICLES IN A
CERTAIN AREA
INTRODUCTION OF PARTICLE AR A
RANDOM LOCATION AND RANDOM WALK
OF THE PARTICLE
(BROWNIAN MOTION)
THE PARTICLE ATTACHES TO THE SEED,
WITH A PROBABILITY DEPENDENT ON
STICKING COEDDECIENT OF THE SYSTEM
NEW PARTICLE
INTRODUCED AND ABOVE
STEPS REPEATED
Brownian Motion of a Particle
Some results from previous
simulations
Dendritic Cluster grown in a DLA simulation with 5000 walkers on a 200 X
200 site
Spectral Dimensions for the DLA
model of Colloid Growth,
Paul Meakin, H. Eugene Stanley
REFERENCES
• Diffusion Limited Aggregation a Kinetic Critical Phenomenon
(1981), (T. A. Witten, Jr. and I. M. Sander)
• Model for the growth of electrodeposited ferromagnetic aggregates
under an in-plane magnetic field
(2010) , (C. Cronemberger, L. C. Sampaio, A. P. Guimarães, and
P.Molho)
• Aggregation of Magnetic Microspheres: Experiments and Simulations
(1988) ,(G. Helgesen, ' A. T. Skjeltorp, P. M. Mors, ' R. Botet, and R.
Jullien)
• Magnetization behavior of small particle aggregates
(1998), (K N Trohidou and D Kechrakos)
• Spectral Dimension for Diffusion Limited Aggregate model for colliod
growth, 1983 (Paul Meakin andK N Trohidou and H. Eugene Stanley)
• Scaling Structure of the Surface Layer of Diffusion-Limited Aggregates,
1985 (Thomas C. Halsey, Paul Meakin and Itamar Procaecia)
• Pattern Formation in Diffusion-Limited Aggregation, 1984 (Tamas
Vicsek)
THANKYOU !
QUESTIONS? 