Stochastic numerics for the gas-phase synthesis of
nanoparticles
Shraddha Shekar1 , Alastair J. Smith1 , Markus Sander1 , Markus Kraft1
and Wolfgang Wagner2
1 Department
of Chemical Engineering and Biotechnology
University of Cambridge
2 Weierstrass Institute for Applied Analysis and Stochastics, Berlin
March 2011
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Outline
1 Introduction
Motivation
2 Model
Type space
Particle processes
Algorithm
3 Numerical study
4 Conclusion
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Motivation
What are nanoparticles? Why are they important?
Particles sized between 1-100 nm.
Both inorganic and organic nanoparticles find wide applications in
various fields.
Why model nanoparticle systems?
To optimise industrial operations and to obtain products of highly
specific properties for sensitive applications.
To understand the molecular level properties that are difficult to be
observed experimentally.
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Motivation II
Salient features of the current model:
Fully-coupled multidimensional stochastic population balance model.
Describing various properties of nanoparticles at an unprecedented
level of detail.
Tracking properties not only at macroscopic level but also at a
molecular level.
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Applications of silica nanoparticles
Silica nanoparticles are amorphous and have Si:O = 1:2.
Their applications include:
Catalysis
Bio-medical applications
Support material for functional nanoparticles
Fillers/Binders
Optics
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Physical system
To describe the system at a macroscopic level it is essential to understand
it at a molecular level.
Flame reactor
Silica nanoparticles
P ≥ 1 atm
T ≈ 1100 - 1500 K
Molecular Scale
Industrial Scale
Precursor (TEOS)
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Type space I
Each particle is represented as:
Pq = Pq (p1 , . . . , pn(Pq ) , C).
Particle Pq consists of n(Pq ) primary particles pi with
i ∈ {1, . . . , n(Pq )} and q ∈ {1, . . . , N}, where N is the total number
of particles in the system.
Si
O
Si
Si
O
O O
O
Si
HO
OH
O
O
HO
Si
Si
O
OH
O
pi = pi(ηSi,ηO,ηOH)
Pq = Pq(p1,...,pn(Pq),C)
Figure: Type Space.
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Type space II
Each primary particle pi is represented as:
pi = pi (ηSi , ηO , ηOH )
where ηx (ηx ∈ Z with ηx ≥ 0) is the number of chemical units of
type x ∈ {Si, O, OH}.
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Type space III
C is a lower diagonal matrix of dimension n(Pq ) × n(Pq ) storing the
common surface between two primary particles:

0
···
..
.
..
.

 C21
 .

C(Pq ) =  ..

 C
 i1 · · ·
..
.
···
···
0

0 ···
..
. ···
..
.
Cij
..
. ···
0
..
.
..
.
..
.




.



0
The element Cij of matrix C has the following property:
(
0, if pi and pj are non-neighbouring ,
Cij =
Cij > 0, if pi and pj are neighbouring.
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Particle processes
Particles are transformed by the following processes:
Inception
Surface reaction
Coagulation
Sintering
Intra-particle reaction
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Particle processes
Particles are transformed by the following processes:
Inception
Surface reaction
Coagulation
Sintering
Intra-particle reaction
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Inception
Two molecules in gas phase collide to form a particle consisting of one
primary.
OH
Si OH
HO
OH
+
HO
HO
OH
OH
OH
OH
Si
Si OH HO
Si
Si OHHO
OH
HO
OH HO
-2 H2O
OH
HO
OH
Si
O
OH
[primary particle]
[monomers]
Figure: Inception of primary particles from gas-phase monomers.
An inception event increases the number of particles in the system
molecule + molecule → PN (p1 , C),
C = 0.
Initial state of primary p1 given by:
p1 = p1 (ηSi = 2, ηO = 1, ηOH = 6),
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Inception rate
Inception rate for each particle (Pq ) calculated using the free
molecular kernel:
1
Rinc (Pq ) = K fm NA 2 Cg2 ,
2
NA is Avogadro’s constant, Cg is the gas-phase concentration of the
incepting species (Si(OH)4 ),
s
πkB T 2
fm
K =4
(dg ),
mg
kB is the Boltzmann constant, T is the system temperature, mg and
dg are the mass and diameter respectively of the gas-phase molecule
Si(OH)4 .
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Particle processes
Particles are transformed by the following processes:
Inception
Surface reaction
Coagulation
Sintering
Intra-particle reaction
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Surface reaction
Dehydration reaction between gas-phase monomer and particle
surface:
OH
HO
Si
HO
O
Si
Si
O
+Si(OH)4
O
-H2O
Si
Si
O
O
OH
O
O
O
O
Figure: Surface reaction between a particle and a gas-phase molecule.
Surface reaction transforms particle as:
Pq + molecule → Pq (p1 , ., pi 0 , .., pn(Pq ) , C0 ),
pi0 → pi (ηSi + 1, ηO + 1, ηOH + 2).
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Particle rounding due to surface reaction
Surface reaction also alters the common surface (C → C0 ).
Net common surface area of pi changes due to volume addition:
∆s(pi ) = (v (pi 0 ) − v (pi ))
2σ
,
dp (pi )
where σ is the surface smoothing factor (0 ≤ σ ≤ 2).
C0 is given by:
(
Cij0 =
0, if pi and pj are non-neighbouring ,
Cij + ∆s(pi ), if pi and pj are neighbouring.
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Surface reaction rate
Surface reaction rate calculated using equation of Arrhenius form:
Ea
Rsurf (Pq ) = Asurf exp −
ηOH (Pq )NA Cg ,
RT
Asurf is pre-exponential factor (obtained from collision theory),
Ea is activation energy,
ηOH (Pq ) is the total number of –OH sites on particle Pq .
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Particle processes
Particles are transformed by the following processes:
Inception
Surface reaction
Coagulation
Sintering
Intra-particle reaction
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Coagulation
Two particles collide and stick to each other:
Pr
Pq
Ps
+
Figure: Coagulation between two particles.
Coagulation of particles Pq and Pr forms new particle Ps as:
Pq + Pr → Ps (p1 , ..., pn(Pq ) , p(n(Pq )+1) , ..., pn(Pq )+n(Pr ) , C).
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Coagulation II
Primary pi from particle Pq and primary pj from Pr are assumed to be
in point contact.
The matrix C(Ps ) is calculated as:

..
.
Cij
..
.




C(Pq ) · · ·
· · ·




..
C(Ps ) = 

.



. . .
Cji
. . . C(Pr )


..
.
and has dimension n(Ps ) × n(Ps ), where n(Ps ) = n(Pq ) + n(Pr ).
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Coagulation rate
Coagulation rate between Pq and Pr calculated using transition
coagulation kernel:
K tr (Pq , Pr ) =
K sf (Pq , Pr )K fm (Pq , Pr )
,
K sf (Pq , Pr ) + K fm (Pq , Pr )
where the slip-flow kernel is:
2kB T 1 + 1.257Kn(Pq ) 1 + 1.257Kn(Pr )
K sf (Pq , Pr ) =
+
3µ
dc (Pq )
dc (Pr )
× (dc (Pq ) + dc (Pr )) ,
and the free molecular collision kernel is:
r
1
2
πkB T
1
1
fm
(dc (Pq ) + dc (Pr ))2
K (Pq , Pr ) = 2.2
+
2
m(Pq ) m(Pr )
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Particle processes
Particles are transformed by the following processes:
Inception
Surface reaction
Coagulation
Sintering
Intra-particle reaction
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Sintering
Sintering described using viscous-flow model.
pi
pj
No Sintering
pi
pk
pj
Partial Sintering
Complete Sintering
Figure: Evolution of sintering process with time.
Sintering between pi and pj of a single particle Pq calculated on a
primary particle-level.
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Sintering level
Sintering level defined to represent degree of sintering between pi and
pj :
1
Ssph (pi ,pj )
− 23
Cij
.
s(pi , pj ) =
1
1 − 23
Ssph (pi , pj ) is the surface area of a sphere with the same volume as
the two primaries.
Pq conditionally changes depending on the sintering level s(pi , pj ).
Two types are defined depending upon a threshold (95%):
Partial sintering s(pi , pj ) < 0.95
Complete sintering s(pi , pj ) ≥ 0.95
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Partial sintering
Surface areas of primaries are reduced by a finite amount.
–OH sites at contact surface react to form Si–O–Si bonds:
pj
pi
OH OH
OH
HO
OH
OH OH OH
OH
pi
OH OH
OH
OH
OH
OH HO
Si
O
Si
Si
O
O Si
OH
Si O
O OH HO O
OH
Si
Si OH
OH
OH OH
OH
Si
Reactions at particle neck
OH
OH
OH
O
Si
Si
Si
O O
Si
O Si
OH
Si O
O O
O
Si
OH
Si OH
OH
OH OH
OH
-2H2O
Figure: Dehydration reaction due to sintering.
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OH
OH
OH
OH
HO
OH
OH
OH
pj
OH OH OH
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OH
OH
OH
OH
Partial sintering II
Surface density of –OH sites assumed constant throughout sintering.
The change in the internal variables of primaries pi and pj given by:
∆ηOH (pi ) = ∆ηOH (pj ) = ρs (Pq )∆Cij /2,
∆ηO (pi ) = ∆ηO (pj ) = −0.5 × ∆ηOH (pi ),
∆ηSi (pi ) = ∆ηSi (pj ) = 0.
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Partial sintering III
Particle continuously transforms due to partial sintering as:
Pq (p1 , . . . , pn(Pq ) , C) → Pq (p1 , .., pi0 , pj0 , .., pn(Pq ) , C0 )
where,
pi0 = pi (ηSi , ηO − ∆ηO (pi ), ηOH − ∆ηOH (pi )),
pj0 = pj (ηSi , ηO − ∆ηO (pj ), ηOH − ∆ηOH (pj )).
Element of C0 given by:
Cij0 = Cij −
∆t
(Cij − Ssph (pi , pj )) ,
τ (pi , pj )
where ∆t is a time interval.
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Complete sintering
pi and pj replaced by new primary pk00 .
Particle transforms due to complete sintering as:
Pq (p1 , . . . , pn(Pq ) , C) → Pq (p1 , .., pk00 , .., pn(Pq ) , C00 ),
where new primary:
pk00 = pk00 (ηSi (pi ) + ηSi (pj ), ηO (pi ) + ηO (pj ), ηOH (pi ) + ηOH (pj )) .
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Complete sintering II
C is changed by removing columns and rows i and j and adding new
column and row k:


0
···
0
··· 0
..
..
..
.. 

..

.
.
.
.
. 


00
00

Ck1
···
0
··· 0 
C =
.
..
..
..
.. 

..
. . 

.
.
.
00
C(n(Pq )−1)1 · · · C(n(Pq )−1)k · · · 0
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Sintering rate
Sintering rate between pi and pj equivalent to rate of change of their
common surface ∆Cij in time ∆t:
∆Cij
1
=−
(Cij − Ssph (pi , pj )),
∆t
τ (pi , pj )
Ssph (pi , pj ) is the surface area of a sphere with the same volume as
the two primaries.
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Characteristic sintering time
Characteristic sintering time of pi and pj is:
dp,crit
Es
τ (pi , pj ) = As × dp (pi , pj ) × exp
1−
,
T
dp (pi , pj )
where dp (pi , pj ) is the minimum diameter of pi and pj , and
As , Es and dp,crit are sintering parameters.
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Particle processes
Particles are transformed by the following processes:
Inception
Surface reaction
Coagulation
Sintering
Intra-particle reaction
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Intra-particle reaction
Reaction of two adjacent –OH sites on one particle:
OH Si HO
O
O
O
Si
O
O
Si
O
- H2 O
Si
O
O
O
Si
Si
O
O
Figure: Intra-particle reaction.
Intra-particle reaction transforms particle as:
Pq (p1 , .., pi , .., pn(Pq ) , C) → Pq (p1 , .., pi0 , .., pn(Pq ) , C),
where
pi0 → pi (ηSi , ηO + 1, ηOH − 2).
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Intra-particle reaction rate
Deduce rate of intra-particle reaction from surface reaction rate and
avergae sintering rate such that Si/O ratio of 1/2 is attained:
Ea
Rint (Pq ) = Asurf exp −
ηOH (Pq )NA Cg
RT


n(Pq )
ρs (Pq )  X Cij − Ssph (pi , pj ) 
−
,
2
τ (pi , pj )
i,j=1
where ρs (Pq ) = ηOH (Pq )/S(Pq ) is the surface density of active sites.
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Algorithm
Input: State of the system Q0 at initial time t0 and final time tf .
Output: State of the system Q at final time tf .
t ←− t0 ,Q ←− Q0 ;
while t < tf do
Calculate an exponentially distributed waiting time τ with parameter;
Rtot (Q) = Rinc (Q) + Rcoag (Q) + Rsurf (Q) + Rint (Q).
Choose a process m according to the probability
P(m) =
Rm (Q)
,
Rtot (Q)
where Rm is the rate of the process m ∈ {inc, coag, surf, int};
Perform process m;
Update sintering level of all particles;
Increment t ←− t + τ ;
end
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Numerical study
For a given property of the system ξ calculated using Nsp
computational particles and L number of independent runs, the
empirical mean is:
L
(Nsp ,L)
µ1
(t) =
1 X (Nsp ,l)
ξ
(t).
L
l=1
The empirical variance is:
L
(N ,L)
µ2 sp (t)
2
1 X (Nsp ,l) 2
(N ,L)
=
ξ
(t) − µ1 sp (t) .
L
l=1
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Confidence interval
The confidence interval IP within which there is a probability P of
finding the true solution is then given by:
h
i
(N ,L)
(N ,L)
IP = µ1 sp (t) − cP , µ1 sp (t) + cP .
s
c P = aP
(Nsp ,L)
µ2
L
(t)
.
We use aP = 3.29 which corresponds to P = 0.999(99.9%)
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Error
The error e is:
(N ,L)
e (Nsp ,L) (t) = µ1 sp (t) − ζ ∞ (t) ,
ζ ∞ (t) is an approximation for the true solution which is obtained from
a "high-precision calculation" with a very large number of particles.
The average error over the entire simulation time is:
M
1 X (Nsp ,L)
ē(Nsp , L) =
e
(tj ),
M
j=1
where the M time steps tj are equidistant.
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M0
The zeroth moment is the particle number density:
N(t)
M0 (t) =
Vsmpl
12
9.5
10
M0
11
10
N =64, L=256
sp
M0 (cm−3 )
log(ē) (cm−3 )
9
8.5
sp
Nsp=16384, L=4
10
10
High Precision Solution
9
8
7.5
1.5
N =512, L=128
10
8
2
2.5
3
3.5
4
4.5
5
10
0
0.1
0.2
log(Nsp )
0.3
0.4
0.5
0.6
0.7
0.8
Time (s)
Figure: Convergence of zeroth moment (Nsp × L = 65536). Solid line indicates
slope of -1.
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Volume
The average particle volume:
V (t) =
1
N(t)
Σq=1 V (Pq (t))
N(t)
−9
−17.6
7
x 10
6
−17.8
Average Volume
Volume (cm3 )
log(ē) (cm3 )
5
−18
−18.2
−18.4
4
N =64, L=256
sp
3
N =512, L=128
sp
Nsp=16384, L=4
2
−18.6
−18.8
1.5
High Precision Solution
1
2
2.5
3
3.5
4
4.5
5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Time (s)
log(Nsp )
Figure: Convergence of average volume (Nsp × L = 65536). Solid line indicates
slope of -1.
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Collision diameter
The average collision diameter of a particle:
1
N(t)
Σq=1 dc (Pq (t))
Dc (t) =
N(t)
1
80
70
Collision Diameter (nm)
0.8
Average Collision Diameter
log(ē) (nm)
0.6
0.4
0.2
0
−0.2
−0.4
1.5
60
50
40
30
Nsp=64, L=256
Nsp=512, L=128
20
Nsp=16384, L=4
10
2
2.5
3
3.5
4
4.5
5
0
0
log(Nsp )
High Precision Solution
0.2
0.4
0.6
0.8
Time (s)
Figure: Convergence of average average collision diameter (Nsp × L = 65536).
Solid line indicates slope of -1.
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Computational time
3
10
80
2
10
Collision Diameter (nm)
Computational Time (s)
70
Nsp=64
Nsp=512
Nsp=163824
High Precision Solution
1
10
60
50
40
30
Nsp=64, L=256
Nsp=512, L=128
20
Nsp=163824, L=4
High Precision Solution
10
0
10 1
10
2
10
3
10
4
10
5
10
0
0
0.1
0.2
0.3
0.4
Figure: Computational time for different Nsp .
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Time (s)
Nsp
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0.6
0.7
0.8
Computer-generated TEM
Figure: TEM images generated by projecting particles onto a plane. Experimental
values from Seto et al. (1995).
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Conclusion
Description of a detailed population balance model.
Numerical studies performed.
Demonstrated feasibility of using first-principles to model complex
nanoparticle synthesis processes.
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Thank You!
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