Dla

MONTE CARLO SIMULATION to study surface diffusion and reaction processes on a fractal catalyst

-Diffusion Limited Aggregation (DLA)

Rajarshi Guha(07302009) Ankit Sharma (07302011) Rahul Kumar (07302006) M.Tech. 2nd Year CL 676 course project

What is a fractal • A fractal is a shape made of parts similar to the whole in some way. It is characterized by self-similarity. • Benoit Mandelbrot lists two qualities that are frequently associated with fractals:  Invariance under displacement: under motion different regions of the object must look similar to each other.  Invariance under scaling: selfsimilarity. Take a large DLA-cluster and simplify it's contour, scale it down to the size of a small DLA-cluster, they will look quite similar.

What is DLA • Diffusion Limited Aggregation (DLA) occurs when a particle undergoes Brownian motion until it makes contact with and sticks to a free-floating cluster of particles. • DLA- “a kinetic critical phenomenon” was invented by two physicists, T.A. Witten and L.M. Sander in 1981. • There are 3 methods of fractal growth1. Percolation 2. Particle-Cluster Aggregation (PCA) 3. Cluster-Cluster Aggregation (CCA) • DLA comes under the category of PCA.

Applications of DLA Physical Processes: • Electrochemical Deposition • Viscous Fingering • Dielectric Breakdown • Monolayer formation on surface (Catalyst formation, carbon deposits etc.) • Polymer generation from solution • Protein Folding etc. Natural Processes: • Snow Flakes • Lichen, Fern and Pine growth • Coral Growth • Creation of planets • Lightning • Corneal neovuscularization etc.

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Real World Phenomena: • Geographical boundary of a country • Rural to urban migration and continuous growth • Web page and computer game design etc.

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Simulation Generated Christmas Tree

Applications Continued..

Applications continued (simple computer simulations)..

Laws of DLA Fractal growth laws: Power law: If N(r) denotes the number of particles within radius r of a circle (or sphere) encircling a DLA then d N(r)= k r For cluster in a plane d = dm ~ 1.71 is the Fractal dimension. Density-Density correlation: The density-density correlation is defined by Where N is the number of particles and p(r)=1 when lattice node is occupied and p(r)=0 when empty. Power law correlation: C(r) and r in a special range (l
Laws of DLA continues..

Mean square displacement (<X²(t)>)Power law: When diffusion occurs on the ideal surface, the diffusion follows Einstein relation given by in one dimensionA growth process is called diffusion-limited when the aggregate increases in size by one particle at a time rather than by bunches. This happens since the density of particles is low and thus the particles do not come into contact with each other before reaching the aggregate. On a fractal surface Einstein relation does not hold. A power law type correlation can be written asWhere d w >2 is anomalous (or walk) diffusion exponent.

DLA Characteristics Lacunarity effect:

• Lacunarity is a counterpart to the fractal dimension, and describes the degree of gappiness of a fractal. • A fractal is very lacunar if its holes tend to be large. • Recent analysis by Benoit Mandelbrot et al. of very large simulated clusters (more than millions of particles) seem to show, that the cluster becomes more compact while growing.

Radius of Gyration :

The fractal dimension D for a cluster can be defined by the expression

where rg is the radius of gyration for the cluster and N(rg) is the total number of particles within that cluster.

DLA Characteristics continues..

Sticking Coefficient: • Stickiness describes the probability that if a particle strikes the cluster, what is the chance of getting added.

S=0.05

• Stickiness=1 => Wandering particle struck part of the existing structure it always stuck.

S=0.01

Symmetric DLA – Simulation Results1 200 180

Iterations =10000

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Number of Particles = 3789, 200 x 200 grid, Seed at the center, Sfactor= 3.

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Density-Density Correlation Function

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Symmetric DLA – Simulation Results2 200

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Number of Particles = 21234, 200 x 200 grid, Seed at the center, Sfactor= 3.

Simulation Continues... 1200

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Asymmetric DLA – Simulation Results1

5 Seeds Asymmetric growth from seeds

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Asymmetric DLA-Simulation Results2 2.5

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Analysis of asymmetric fractals generated 1

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Fractal Dimension = 2-0.25=1.75 0.1

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Brownian Motion <X²(t)>=88.13 x t^(0.87) dw = 2.29

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Analysis of symmetric fractals generated 1200 200 180

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Analyzing by curve fitting we geta = 0.35. hence fractal dimension = 2-0.35 = 1.65

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Power correlation law: C(r)=2.64 x r^(-0.35)

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Mean square Displacement: <X²(t)>=207 x t^(0.41) dw = 4.88

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Simulation Procedure (Algorithm) Symmetric DLA

Asymmetric DLA

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Initialize the grid and define necessary variables (eg. Define diffusion time, seed etc.)

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Initialize grid, define necessary variables and distribute seeds randomly.

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Release particles from the periphery of the circle.

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Apply Brownian motion until it finds the cluster.

Generate a particle within grid and put it in random walk until a particle is encountered.

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It sits beside the encountered particle.

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The process continues until definite number of particles is attained.

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Keep on tracking distance from seed and kill particle if it goes outside.

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If particle satisfies Sfactor criteria it is allowed to proceed.

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Then particle adheres to one of the free available sites.

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The process goes on until the grid circumference is reached.

Application N2O decomposition kinetics by MC simulation (Guo et al., 1994): 2 step kinetics :

Surface coverage θ, when R1=R2 can be expressed as Theoretically: For the 2 step reaction mechanism 2nd step occurs on Catalyst surface. Rate constant K2 is related to speed of surface diffusion v2D by-

Speed of surface diffusion is directly proportional to surface diffusion coefficient. Hence, At constant temperature K2 ∞ D. This approach gives K2 = 0.6

Application Continues.. Ideal surface: Overall rate law by MC simulation Fractal Surface:

Figure showing the DLA fractal generated By Guo et al. (Fractal dimension 1.72) Monte Carlo simulation generated K2 = 0.58 Theoretical K2 = 0.6

Conclusions • Many physiconatural processes are random and MC DLA simulation can actually represent those processes computationally.

• DLA fractals are recognized by their fractal dimension and power law correlation.

• An important application of DLA is to simulate fractal catalyst surface and thereby modifying the rate law of surface reaction.