The process of nanoparticle agglomeration as a function of the monomer-monomer interaction potential is simulated numerically by solving Langevin equations for a set of interacting monomers in three dimensions. The simulation results are used to investigate the structure of the generated clusters and the collision frequency between small clusters. Cluster restructuring is also observed and discussed. We identify a time-dependent fractal dimension whose evolution is linked to the kinetics of two cluster populations. The absence of screening in the Langevin equations is discussed and its effect on cluster translational and rotational properties is quantified.
AWS Community Day CPH - Three problems of Terraform
Talk given at the Particle Technology Lab, Zurich, Switzerland, November 2008.
1. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions
Nanoparticle collisional dynamics by
Langevin simulations
Lorenzo Isella and Yannis Drossinos
Joint Research Centre, Ispra, Italy
ETH, November 2008
EC DG JRC – TFEIP - November 2006
2. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions
Problem Formulation
Motivation and Goals
Simulation of soot particle agglomeration via Langevin
equations.
Aggregate static properties: radius of gyration,
hydrodynamic radius, fractal dimension, coordination
number
And dynamic properties: transport (diffusion coefficient),
response time, thermalization.
Agglomeration dynamics and numerical evaluation of the
collisional kernel matrix elements, comparison with
Smoluchowski kernel.
EC DG JRC – TFEIP - November 2006
3. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions
Model for Monomer Dynamics
Langevin Equation for Mesoscopic Systems 1/2
3D system of interacting monomers, each obeying
m1¨i = Fi − β1 m1 ri + Wi (t).
˙
r
Force acting on i-th monomer from pairwise
monomer-monomer interaction potential
1
Fi = − ri Ui = − ri u(rij ) .
2
j=i
White noise acting on each monomer
Wij (t) = 0 Wij (t)Wij (t ) = Γδii δjj δ(t − t ),
and
noise strength Γ = 2β1 m1 kB T fixed by fluctuation-
dissipation theorem.
EC DG JRC – TFEIP - November 2006
4. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions
Model for Monomer Dynamics
Langevin Equation for Mesoscopic Systems 2/2
MD ⇒ microscopic description of the system. Langevin
thermostat (among many) to model the coupling of the
system with a thermal bath and define temperature for the
system.
Nanoparticles ⇒ mesoscopic description of the system.
Langevin equation as coarse-grained description of the
nanoparticle dynamics. Noise term accounting for the
effect of fluid-molecule-to-nanoparticle collisions giving rise
to nanoparticle diffusion.
Simulations performed with a MD package (ESPResSo)
but results interpreted for nanoparticles.
EC DG JRC – TFEIP - November 2006
5. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions
Model for Monomer Dynamics
General considerations
Langevin equation for interacting monomers ⇒ cluster
properties and dynamics fixed by the monomer-monomer
interaction potential only.
A cluster of monomers is not a primitive concept; only
monomer properties are specified in the model hence
Cluster fractal dimension, coordination number, friction
coefficient, collisional kernel etc. . . are a model output.
Langevin equation does not include monomer screening in
a cluster ⇒ effects on cluster mobility.
EC DG JRC – TFEIP - November 2006
6. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions
Dimensionless Formalism
Specification of the Units
Natural (but not unique!) choice for time, distance and
mass units
t ≡ τ1˜
r ≡ σ˜, ˜
m1 ≡ m1 m1 .
r t,
Temperature unit T ∗ is a derived quantity. For a 20nm soot
particle (ρp 1.3g/cm3 ) in air at room temperature
182 πµ2 σ
m1 σ 2
T∗ = f
650K
=
2 6kB ρp
kB τmon
˜
⇒ T ≡ T /T ∗ = 0.5 for exhaust nanoparticles at room
temperature.
Dimensionless quantities used in the following, unless
EC DG JRC – TFEIP -stated. 2006
otherwise November
7. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions
Interaction Potential
Features of Monomer-Monomer Interaction Potential
Repulsion at short separations r ≤ σ (hard-core repulsion)
and attraction for separations above σ (sticking upon
collision).
Simulations performed with two radial interaction
potentials: integrated Lennard-Jones potential (model for
the attractive part of Van der Waals interaction between
two spheres, ∼ r −6 for r σ) and with a short-ranged
model potential.
Model Potential
Van der Waals
50
0
()
u(r)
−50
−100
1.0 1.1 1.2 1.3 1.4
EC DG JRC – TFEIP - November 2006 r
8. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions
Interaction Potential
Non-coalescent monomers
Hard-core repulsion: monomers do not compenetrate but
retain their identity after colliding ⇒ no coalescence.
In ESPResSo, a “softer” monomer-monomer interaction
potential leads to overlapping but not coalescing
monomers. No primitive concept of monomer radius, only
of monomer mass and monomer-monomer interaction
potential.
m1 m1 2m1
2m1
m1 m1
EC DG JRC – TFEIP - November 2006
9. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions
Numerical Implementation
Overview of the Numerical Simulations
5000 monomers placed randomly in a box with periodic
boundary conditions and density ρ = 0.01.
Mitigate the role of initial conditions ⇒ results averaged
over 10 simulations.
At the end of the simulation, the aggregate concentration is
almost two orders of magnitude smaller than initially.
MD ESPResSo package to solve the 3D Langevin
equations (Verlet algorithm and Euler scheme for
evaluating stochastic force on monomers).
Unlike early studies (Meakin, Mountain), not necessary to
look for agglomeration events while evolving the system.
Each monomer can be addressed individually at all times
(one can “label” it).
Only monomer positions and velocities are returned ⇒
how to identify the clusters?
EC DG JRC – TFEIP - November 2006
10. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions
Post-Processing
Distances and Graphs
Distance between i-th and j-th monomer along e.g. x-axis
in a periodic box of size L
xi − xj
(x)
= xi − xj − L · nint ≤ L/2
Dij
L
Total distance between i-th and j-th monomer
(x) 2 (y ) 2 (z) 2
Dij = Dij + Dij + Dij
Fix a distance dthr and calculate symmetric adjacency
matrix
1, if Dij ≤ dthr
Aij =
0, otherwise
In graph theory, a symmetric Aij identifies completely a
non-directed graph.
What does it have to do with clusters?
EC DG JRC – TFEIP - November 2006
11. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions
Post-Processing
Cluster Detection: Clusters as Graphs
C
A
B
1
2 4
6 1
3
5
5 3 6
4
2
i-th and j-th monomer bound together (adjacent) if their
distance Dij ≤ dthr , dthr close to req 1.02σ.
Any monomer in a cluster can be reached from any other
monomer in the same cluster by strides of length dthr .
For a fixed dthr , cluster determination from monomer
positions ⇔ determination of connected components of a
non-directed graph.
Each monomer configuration (hence each cluster) has a
unique representation as a graph via the adjacency matrix
Aij i,j 2 4 5
C
2 1 1 0
Aij :
2 4
4 1 1 1
5
EC DG JRC – TFEIP - November 2006 5 0 1 1
12. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions
Post-Processing
Collision Statistic: Clusters as Sets of Monomers
t + δt
t
9
1
1
2 2
4
6
9
4 7
5
6
7 10
3
8
3
10
5
8
Clusters as not ordered collections of monomers (no
monomer label is repeated) ⇒ mathematical definition of a
set.
If two independent clusters at time t become a proper
subset of the same cluster at time t + δt ⇔ collision.
Fragmentation as the reverse process of a sticky collision.
Kernel elements βij from i and j-mer concentrations and
from number of collisions per unit volume in δt
Nij /δt = (2 − δij )βij ni nj .
EC DG JRC – TFEIP - November 2006
13. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions
Determination of the Fractal Dimension
EC DG JRC – TFEIP - November 2006
14. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions
Determination of the Fractal Dimension
Distribution of Aggregate Morphologies
EC DG JRC – TFEIP - November 2006
15. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions
Determination of the Fractal Dimension
Time-averaged Fractal Dimension
(ri − rCM )2
2 2
Radius of gyration Rg = i + rp .
k
Time-independent df from average Rg = αk 1/df from k = 5
(power-law breaks down for smaller k ).
Two slopes ⇒ DLA (df 2.4 − 2.5) and cluster-cluster
(df 1.7 − 1.8) aggregation.
10.0
Small clusters k ≤ 15
small
Rg~k1 df , for 4<k ≤ 15
Large clusters k >15
large
Rg~k1 df
5.0
for large clusters
Rg~k1 df
for k ≥ 5
large
df =1.56
2.0
df=1.62
Rg
dsmall=2.25
1.0
f
0.5
1 2 5 10 20 50 100 200 500
EC DG JRC – TFEIP - November 2006 k
16. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions
Determination of the Fractal Dimension
Time-dependent Fractal Dimension
t
From ensemble data at time t, fit Rg = αk 1/df .
Evolution of dft determined by kinetics of large and small
cluster populations.
large
Fractal dimension decreases and tends to df .
2.2
2.0
dtf
1.8
1.6
0 500 1000 1500 2000 2500 3000
Time
2 10 60 500 5000
All clusters
Small clusters k ≤ 15
Large clusters k >15
N∞Vbox
0 500 1000 1500 2000 2500 3000
EC DG JRC – TFEIP - November 2006 Time
17. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions
Cluster structure
Cluster coordination number
Coordination number calculated from cluster Aij .
6
5
Mean coordination number
4
3
2
1
0
0 500 1000 1500 2000 2500 3000
Time
High coordination number ⇒ low df from elongated shape
of large aggregates, not from cluster cavities.
df by itself insufficient to characterise cluster morphology.
High coordination number as intrinsic feature of Langevin
EC DG JRC – TFEIP - November 2006Araki, Phys. Rev. Lett., 85).
simulations (Tanaka &
18. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions
Cluster structure
Cluster restructuring
Radial monomer-monomer interaction potential ⇒ locking
of relative distance between neighbouring monomers but
bonds can move on the monomer surface.
2.8
2.6
2.4
2.2
2.8
Rg
2.0
Rg
2.1
1.8
1.4
1.6
550 575 600
Time
1.4
500 600 700 800 900 1000
Time
Restructuring on a time-scale ∼ τ1 after collision.
Aggregate restructuring from Langevin equation and radial
interaction only.
EC DG JRC – TFEIP - November 2006
19. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions
Transport Properties and Response Time
Translational diffusion coefficient 1/2
25
0.5
−1
10
20
−5
2 × 10
15
<δr2 >
δ CM
0.1 5 15
1
10
Simulation
Linear fit
Power−law fit
5
0
0 100 200 300 400
Time
Ensemble average on 800 trajectories (k = 50 above)
2
δrCM (t) kB T
t→∞
Dk − −
−→ Cs (Kn).
=
6t k m1 βk
Consistent with Cs (Kn) = 1 and βk = β1 ⇒ continuum
regime and cluster relaxation time identical to τ1 .
2 t γ , γ 3 at early times ⇒ Brownian particle.
δrCM (t)
EC DG JRC – TFEIP - November 2006
20. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions
Transport Properties and Response Time
Translational diffusion coefficient 2/2
k -mer diffusion coefficient Dk ∝ 1/k hence
Dk does not “see” cluster structure, only its size k
Cluster mobility radius Rm (radius of a sphere with the
same diffusion coefficient) defined via
kB T 1
∝.
Dk =
6πµf Rm k
Rg = αk 1/df , whereas Rm = kr1 ⇒ Rm Rg for large
clusters.
Effect of introduction of a shielding factor for the i-th
monomer in Langevin equation
0 (perfect shielding) ≤ ηi ≤ 1 (no shielding).
EC DG JRC – TFEIP - November 2006
21. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions
Transport Properties and Response Time
Numerical experiment: adjusting friction and noise strength 1/2
Langevin equation with monomer shielding factor ηi
m1¨i = Fi − ηi β1 m1 ri + 2 ηi β1 m1 kB T δt N(0, 1).
˙
r
Heuristic expression ηi = 1 − neighbours(i−th monomer) /12 .
Fluctuation-dissipation theorem (FDT) enforced for each
monomer, not for the whole aggregate.
12
η=1
η= heuristic expression
10
8
<δr2 >
δ CM
6
4
2
0
0 20 40 60 80 100
EC DG JRC – TFEIP - November 2006 Time
22. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions
Transport Properties and Response Time
Numerical experiment: adjusting friction and noise strength 2/2
The {ηi }’s enhance cluster translational diffusion ⇒ they
tend to close the gap between Rm and Rg .
Understandable enhancement: from FDT a Brownian
particle with friction ηβ1 has diffusion coefficient
(η) (η=1) (η=1)
D1 = kB T /(m1 ηβ1 ) = D1 .
/η≥D1
Assuming FDT, cluster friction is obtained from diffusion
{η }
simulations: βk = kB T /(km1 Dk i ).
Does FDT hold for a cluster? Numerical investigation of
time-decay of cluster initial velocity v?
How to choose the {ηi }’s? Monomer exposed surface?
EC DG JRC – TFEIP - November 2006
23. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions
Transport Properties and Response Time
Cluster rotation 1/2
Rotations in 3D identified by 3 Euler angles representing
the spatial orientation of any frame as a composition of
rotations from a reference frame
Cluster as a rigid body: motion fixed by the coordinates of
three non-collinear monomers rA (t), rB (t) and rC (t).
3x3 rotation matrix A from X = [rA (0), rb (0), rc (0)] and
X = [rA (t), rb (t), rc (t)]
−1
T
XXT
A=XX .
EC DG JRC – TFEIP - November 2006
24. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions
Transport Properties and Response Time
Cluster rotation 2/2
For random√rotation matrices, α and γ uniform in
√
δα2 = δγ 2 = π/ 3 1.81, whereas
[−π, π] ⇒
δβ 2
β = arccos(1 − 2U(0, 1)) − π/2 ⇒ 0.68.
2.0
Angle standard deviation (rad)
1.5
< δα2 >
10
< δβ10 >
2
< δγ2 >
10
1.0
< δβ50 >
2
0.5
0.0
0 100 200 300 400
Time
No preferential cluster rotation angle/orientation; an initially
ordered ensemble of large clusters takes longer to reach
random orientation.
EC DG JRC – TFEIP - November 2006
25. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions
Agglomeration Dynamics
Analytical Expressions for the Collisional Kernel
Smoluchowski kernel
2kB T 1/df
i −1/df + j −1/df .
Sm
+ j 1/df
i
βij =
3µf
Diffusion coefficient from Langevin simulations and
non-continuum effects in aggregate collisions (Ri ≡ Rg,i )
4πkB T −1
i + j −1 Ri + Rj βF (Ri , Rj ).
LD
βij =
m1 β1
data from simulations i < 5
where small
αsmall i 1/df , if 5 ≤ i ≤ 15
Ri =
α i 1/dflarge , if i > 15
large
Kernel homogeneity exponent λ:
βγi,γj = γ λ βi,j ⇒ N∞ ∼ t −1/(1−λ) .
βij : λ = 0 ⇒ N∞ ∼ t −1 ;
Sm
large
) − 1 ⇒ N∞ ∼ t −0.72 .
LD
βij : λ = (1/df
EC DG JRC – TFEIP - November 2006
27. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions
Agglomeration Dynamics
Evaluation of the kernel elements βij
1.2 × 10−52.4 × 10−5
q
Simulations
q q
q
β
q
2βijninj
q
q
q
q q
N13/δt q
δ q
q
q
q
q
q
q
q
q
q qq
qq
qq
qqq
qqqq
q qqqqqq
qqqqqqqqqqqq
q
0
0 20 40 60 80 100
Time
0.5
1j (numerical)
βij/(8kBT/3µf)
1j (analytical)
µ
2j (numerical)
0.35
2j (analytical)
0.2
3
1 2 4
j
βij numerically determined by fitting Nij /δt from simulations
to (2 − δij )βij ni nj at all times.
Calculations for i, j = 1 . . . 4 but not enough data for the
other kernel elements.
EC DG JRC – TFEIP - November 2006
28. Introduction Physical Scales and Interaction Potential Simulations and Post-Processing Results and Discussion Conclusions
Final Remarks
Use MD techniques to investigate aggregate collisional
dynamics as a function of the monomer-monomer
interaction potential.
Cluster identification by considering them as connected
components of a graph.
Time-dependent dft linked to the kinetics of two cluster
populations.
Diffusion coefficient of a k -mer scaling like k −1 ⇒
aggregates are “transparent” to the fluid and with the same
response time of a monomer.
Investigation of cluster rotational properties.
Numerical calculation of the kernel elements βij for low
indexes.
EC DG JRC – TFEIP - November 2006
