Monte Carlo Case Study

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Monte Carlo Case Study

• Surface to Surface Particle Transport
• by
• Tiffany McKnight
• Yuan Ji
• Gang Chen

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Brief Review

• Monte Carlo method provides approximate solutions
  to problems using statistical sampling
• Uses N-point evaluations and M-dimensions

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Brief Review Cont.

• What the method needs
• a physical or mathematical system
• this system needs to be described by a number of
  pdfs
• probability density functions
• many trials are done
• the result is the average over the number of
  observations

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Brief Review cont.

• Some Examples
• Using a simulated dart board to determine PI
• Using a simulation to determine the movement of a
  beam of radiation through a cancer patient
• an example of Surface to Surface Particle
  Transport

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History of the Monte Carlo Method

• The method is named after the city of Monte Carlo
• because of the roulette wheel being a simple
  random number generator
• This method dates back to about 1944
• although there were instances of people using
  random chances to determine PI and other things

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History cont.

• This name was coined by Metropolis during the
  Manhattan Project
• Fermi, Metropolis, and Ulam obtained estimates of
  the Schrodinger equation using the Monte Carlo
  method
• One of the first times it was used as discovery
  and not just data verification

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History cont.

• With the advent of computers in the 1970s the
  Monte Carlo method grew in usefulness and
  popularity
• Computational Scientists took advantage of the
  computers speed in computation

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Overview of Surface to Surface Particle Transport

• Trace particle from one surface to another inside
  an enclosure
• complexity is added when a particle interacts
  with the surfaces in an enclosure
• Application areas include
• Thermal Radiative Transport
• Neutron Damage
• Molecular Sputter
• Chemical Vapor Deposition

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Surfaces

• If an object or system is not infinity, it has a
  border. The border could be considered as its
  Surface.
• When two system share a surface, we call the
  surface Interface.
• Usually, when we talk about surface, we mean the
  surface of solid.

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Solid

• There are 2 kinds of solid
• Crystal The basic unit repeats
  periodically. Long range
  periodicity exits.
• Amorphous solid No long range periodicity exist.
  The atoms or
  basic unit can not be repeated periodically.

So, the surface of crystal is deferent from the
surface of amorphous.
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The surface of a crystal could be expressed by
Miller indices For example, if the crystal has
the structure of fcc (face centered cubic)
(001)
(100)
(010)
The surface of amorphous solid could not
expressed by this way.
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Particles on Surface

• Atoms or particles could be absorbed on surface.
  Because of energy distribution, chemical
  potential or other chemical or physical
  processes, they interact with surface. There are
  two kinds of interaction
• Weak Interaction
• Strong Interaction

Particles
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• Weak Interaction
• The particles interact with surface only by
  absorption. If the particles have energy bigger
  than adsorption energy, they could transport to
  other surface.
• The energy between a particle and a surface atom
  could be described as Lennard-Jones type.

Where r0 is the equilibrium distance between two
particles and r is the distance between them, ?
is the minimum value of u(r)
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For a particle adsorbed on the surface, the
adsorption energy is
r 1
r 2
r 3
r 5
r 4
To simply the computation, we could only count
the atoms of surface which are nearest to the
particle.
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Strong Interaction The particles interact with
surface not only by adsorption but also by
chemical bonding. In this case, particles are not
easy to transport/diffuse to other surface. The
activation energy is EsEadEch, where Ead is the
adsorption energy and Ech is bonding energy. To
transport between surfaces, the particle has to
break the chemical bond and absorb an energy
higher than Ead.
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Random Walk
When a particle is diffusing on an surface, it
jumps from one site to the next site. The jumping
direction is random. When we observe a particles
diffusion, we cant know its next position in
advance. This could be described as Random Walk.
1
3
2
rm
4
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In practical problems, because of the exist of
concentration gradient and temperature gradient,
the particle seldom go back to it starting point.
So we can apply a restriction to the random walk
the particles next jump does not cross its
previous jump. This is like the
Self-Avoiding-Walk of a long polymer molecule.
Self-Avoiding-Walk
Non-Self-Avoiding-Walk
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Diffusion Coefficient (Diffusivity)
Diffusion coefficient could be computed by the
following method (Monte Carlo) We could compute
the time t which is needed for particle to jump
from one site to the adjacent site by Quantum
Mechanics. Then, we could generate random
self-avoiding-walks of m steps by hit-or-miss
method of generating random unrestricted walks
and rejecting those that intersect themselves
before the No. m step. So, we got rm . By rm , we
could get
After that, we could repeat the m steps of random
walk N times. If N is big enough, well have
Where v is the square root of variance
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Obviously, the error converges to
zero as N increases.
By now, we could have the value of diffusion
coefficient D
At different temperature, t is different. By
Quantum Mechanics, we could compute a set of
values of t under different temperatures.
According to the equation above, we could get a
set of values of D corresponding to t. Thus, we
have the data of diffusivity (i.e. diffusion
coefficient) D versus Temperature T Then we could
have the relation of D?1/T
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There is relation between diffusivity D and
activation energy Es
Do the logarithm to both sides of the equation
lnD and 1/T are linear to each other, Es is the
slope. By now, we get the value of Es.
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Diffusivity and activation energy are important
parameters for surface to surface transport. In
terms of them we could get the relation between
the flux and the concentration gradient. If we
know the diffusivity and activation energy, we
could control some processes. For example, to
simulate the growth of carbon nanotubes by
chemical vapor deposition (CVD), we have to know
the diffusivity and activation energy of Fe and
carbon diffusing on the surface of the silica.
Carbon Nanotube
Iron particle
Ferrocene and xylene
Silica support
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We also need to know the diffusivity and
activation energy of carbon diffusing on iron
surface. By these parameters, we could compute
the time needed to grow an iron nanoparticles, we
also could compute the time needed to grow a
segment of carbon nanotube. The two time
parameters are important to the research of the
mechanism of the growth of carbon nanotubes. The
computation of diffusivity and activation energy
could reduce the cost of research greatly.
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Monte Carlo Surface to Surface Particle Transport
Application

• - LIGHT TRANSPORT SIMULATION

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What does light transport do?

• - The algorithms used for production work (such
  as scan-line rendering and ray tracing) do not
  have the ability to simulate indirect lighting.
• - If we could find robust light transport
  algorithms, then the indirect lighting could be
  computed automatically, which would make the
  lighting task far easier.

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Monte Carlo Surface to Surface Particle Transport
Application

• Light transport algorithms generate realistic
  images by simulating the emission and scattering
  of light in an artificial environment.
• Applications include lighting design,
  architecture, and computer animation, while
  related engineering disciplines include neutron
  transport transfer.

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Monte Carlo Surface to Surface Particle Transport
Application

• The main challenge with these algorithms is the
  high complexity of the geometric, scattering, and
  illumination models that are typically used.

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Material Properties

• Emission
• the directional distribution of physical
  emissions
• the spatial distribution of emissions over a
  surface
• Particle/surface interactions
• particle is absorbed/emitted

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Some Robust Monte Carlo Methods For Light
Transport Simulation

• Multiple Importance Sampling
• Bi-directional Path Tracing
• Metropolis Light Transport

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Implementation
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Implementation

• many ''standard'' aspects of Monte Carlo are
  simulated some parameters are represented.

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Day lighting Example
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Day lighting Example
Video illustrates the process by displaying
photons as they enter the geometry.
Reference http//csep1.phy.ornl.gov/pt/pt.html
http//graphics.stanford.edu/papers/veach_the
sis/
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