Bivariate Stochastic Modelling of Nanoparticles

1
Bivariate StochasticModelling of Nanoparticles
Neal Morgan1, Clive Wells1, Markus Kraft1,
Wolfgang Wagner21Department of Chemical
EngineeringUniversity of Cambridge2Weierstrass
Institute for Applied Analysis and
StochasticsBerlin
2
Outline of talk

• Introduce the extended model for nanoparticle
  growth.
• Introduce the algorithm to solve the model.
• Simulate a test case
• Simulate two real particle systems
• SiO2 formed in a premixed laminar flame
• TiO2 formed under differing inlet conditions in a
  reactor.

3
Motivation

• The size and shape of particles has a major
  effect on the properties of the material.
• The ability to model the morphology within
  reactors and flames will allow us to determine
  and alter these properties.
• Stochastic methods can recover the full evolution
  of the bivariate particle size distribution over
  time but without the computational expense of
  other numerical techniques.

4
The model
The model used consists of four main processes

• Particle Sintering
• Coagulation
• Particle Inception
• Surface Growth

5
Sintering
Volume v Area a
Volume v Area a0(v/v0)2/3
The difference between the area of the particle
and its theoretical minimum.
The characteristic sintering time. This is often
a function of temperature and particle diameter.
6
Coagulation
Birth term
Death term
bv,v(a,a) is the coagulation kernel that
describes the rate at which particles collide.
In this investigation a free-molecular kernel is
used.
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Particle Inception
From gas phase
Describes the rate at which monomer particles are
created from the gas phase precursor chemicals.
The parameters kg and C are the gas phase rate of
loss of the precursor and the concentration
respectively.
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Surface Growth
From gas phase
Describes the rate at which mass is deposited on
the surface of existing particles in the system.
As with particle inception, the term is dependant
on the gas phase concentration, C and has a rate
for surface growth of ks.
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The Population Balance Model
So how do we solve this?
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Solving the model

• To solve the model we make use of a stochastic
  particle method that simulates the system as an
  ensemble of N particles.
• The mechanisms involved in the simulation are
  assigned rates according the current state of the
  system.
• These rates determine a waiting time.
• The system is advanced by the waiting time and
  the relevant jump performed.
• The system is updated and the procedure repeated.

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The Stochastic Algorithm
Initialize system
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The Test Simulation
The performance of the algorithm was tested using
a simple simulation.

• Coagulation performed using a free-molecular
  kernel at a constant temperature.
• Particle source rate is kept constant
• Surface Growth rate is kept constant

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The Test Simulation
With Surface Growth, Sintering and Coagulation
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The Test Simulation
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Simulated systems

• The algorithm has been used to simulate many
  different nanoparticle systems
• SiO2 in a premixed laminar flame (particle mass
  only)
• Fe2O3 in a premixed laminar flame (particle mass
  only)
• TiO2 in a reactor (particle mass only)
• SiO2 in a premixed laminar flame (particle mass
  and surface area)
• TiO2 in a reactor (particle mass and surface area)

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Simulation of SiO2
The algorithm was used to simulate a low-pressure
1-D premixed laminar flame in which SiO2
particles were being formed.
The flame system is solved using the 1D flame
code PREMIX This code allows us to determine the
parameters needed for the population balance
model.
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Simulation of SiO2
The temperature profile is needed as the
coagulation and sintering terms are temperature
dependant.
1600
1400
1200

• Temperature Profile

1000
Temperature / K
800
600
400
200
0
0.01
0.02
0.03
0.04
0.05
Time / s
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Simulation of SiO2
The rate of production tells us the rate at which
new mass will enter the system.

• Rate of Production
  of
  SiO2

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Simulation of SiO2
Particle size distribution
Average mass and area of particles
0.010
1000
1000
0.008
100
100
0.006
Average mass / m0
Average area / a0
Normalized count
0.004
10
10
0.002
0
1
1
1
1000
100
10
104
0.05
0.04
0.03
0.02
0.01
0
Particle volume / v0
Time / s
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Simulation of SiO2
104
1000
Line of pure coagulation (a v)
Volume Area Correlation
Particle area / a0
100
Line of total coalescence (a v2/3)
10
10
100
1000
104
Particle volume / v0
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Simulation of TiO2
The algorithm simulated a system where TiCl4
oxidizes to TiO2 in a reactor. A simple one step
reaction scheme is used
TiCl4 O2
TiO2 2Cl2 .
The concentration of TiCl4 is assumed to decrease
according to the following equation
where As is the area density of the system.
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Simulation of TiO2
Particle size distributions at various times in
the reactor
Initial concentration of TiCl4 5x10-6 mol/m3,
Temperature 1400 K
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Simulation of TiO2
Average diameter and total concentration at
various times and temperatures
Initial concentration of TiCl4 5x10-6 mol m-3
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Simulation of TiO2
Low initial concentration of TICl4
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Where next?

• The results of the simulations need more
  experimental verification to aid the development
  of the model.
• Additional kinetic mechanisms for the oxidation
  of inorganic species in flames and reactors are
  required for more accurate simulations.
• The algorithm can easily be extended to include
  more internal variables in the population balance
  without incurring any major additional
  computational costs.
• Coupling the algorithm to computational fluid
  mechanics simulations would allow us to resolve
  more complex systems.

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Summary

• Stochastic solution method for the growth and
  morphology of inorganic nanoparticles has been
  introduced.
• The bivariate particle size distributions can be
  obtained very quickly.
• The temperature dependence of sintering could be
  observed in the SiO2 simulations.
• Bi-modality of the TiO2 particle size
  distribution could be observed when particle
  inception was present.
• Surface growth is only important in the studied
  TiO2 system when the initial concentration of
  TiCl4 is high.

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Acknowledgements

• Financial support
• EPSRC (for the financial support of Neal Morgan)
• Oppenheimer Fund (for the financial support of
  Dr. Clive Wells)