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

– 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.

a. Simulated DLA of about 33,000 particles. b. High-voltage dielectric breakdown

c. Copper sulfate in an electro-deposition cell

USING THE DLA MODEL

• 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.

• Fractal Dimension =

)ln(

lnNrD

log(no. of self similar pieces)log(magnification factor)

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:

• Magnetic interactions between two spheres, i and j, separated by the distance ,is given by the following relation,

• We also have two dimensionless parameters, effective strength of dipole-dipole interactions and dipole-field interactions.

)6

(3dM

jiij rrr

23

2 ).)(.(3.

ij

ijjijiji

ijij r

ruruuur

D

TkBK

TkdK

B

extdf

Bdd

3

2

• The total energy of a particle at the position is given by:

• Differently from DLA, the energy difference between the current position and the four possible new positions is used to calculate the probabilities.

• According to this model, the particle moves to the region of lower energy with higher probabilities.

ir

)(.)( iTiimag rBrU

ii

iB

i P

UTkP

)1exp(

EXPERIMENTAL SETUP FOR THE DLA CLUSTER STUDY

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 Procedure

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 simulationFORMATION OF LATTICE AND INTRODUCTION OF

SEED

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

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

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?