experimental and simulated study of diffusion limited
Transcript experimental and simulated study of diffusion limited
EXPERIMENTAL AND SIMULATED
STUDY OF DIFFUSION LIMITED
AGGREGATION OF SUSPENDED
Rabia Aslam Chaudary (12100011)
Aleena Tasneem Khan (12100127)
• Diffusion Limited Aggregation
What is DLA?
The DLA Model and it’s applications
• Our approach to the study:
– Experimental Study
– Simulated study
• Past studies done of DLA clusters
DIFFUSION LIMITED AGGREGATION
What is Diffusion Limited
• 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 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 =
ln( N )
• 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
• 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
• 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
• Diffusion-Limited Aggregation, a Kinetic Critical
(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
(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.
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
Our model for non-magnetic and
• 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
• 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:
• 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
• We also have two dimensionless parameters, effective strength
of dipole-dipole interactions and dipole-field interactions.
d k BT
• The total energy of a particle at the position ri is given by:
( 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.
U i )
• According to this model, the particle moves to the region of
lower energy with higher probabilities.
EXPERIMENTAL SETUP FOR THE DLA
o Study of non-magnetic particles:
Particles doing Brownian motion observed by
microscope and camera. Possibility of cluster
o Study of magnetic particles:
Sulfonated polystyrene magnetic microspheres with 30%
iron oxide dispersed in water confined to a mono-layer.
• Setup to vary temperature
• Application of External Field
External Magnetic field
The effect on Fractal dimensions and scaling properties of the
SIMULATED STUDY OF THE DLA
Outline of simulation
FORMATION OF LATTICE
AND INTRODUCTION OF
LOOP OVER THE DESIRED
NUMBER OF PARTICLES
UNTIL A CLUSTER IS
CALCULATE FRACTAL DIMENSION
BY CALCULATING THE RATIO OF
NUMBER OF PARTICLES IN A
INTRODUCTION OF PARTICLE AR A
RANDOM LOCATION AND RANDOM WALK
OF THE PARTICLE
THE PARTICLE ATTACHES TO THE SEED,
WITH A PROBABILITY DEPENDENT ON
STICKING COEDDECIENT OF THE SYSTEM
INTRODUCED AND ABOVE
Brownian Motion of a Particle
Some results from previous
Dendritic Cluster grown in a DLA simulation with 5000 walkers on a 200 X
Spectral Dimensions for the DLA
model of Colloid Growth,
Paul Meakin, H. Eugene Stanley
• 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
• Aggregation of Magnetic Microspheres: Experiments and Simulations
(1988) ,(G. Helgesen, ' A. T. Skjeltorp, P. M. Mors, ' R. Botet, and R.
• 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