Constant-Number Monte Carlo Simulations

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NSF: EF
-
0830117

Constant
-
Number Monte Carlo Simulations
of Nanoparticles Agglomeration

Yoram
Cohen, Haoyang
Haven
Liu, Sirikarn Surawanvijit,
Robert Rallo and Gerassimos Orkoulas


Center for Environmental Implications of nanotechnology

and

Department of Chemical and Biomolecular Engineering


University of California, Los Angeles


http://www.cein.ucla.edu/

This materials is based on work supported by the National Science Foundation and Environmental Protection Agency under Cooper
ati
ve Agreement
# NSF
-
EF0830117. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(
s) and do not
necessarily reflect the views of the National Science Foundation or the Environmental Protection Agency.

NSF: EF
-
0830117

OUTLINE


Motivation


Toward predictive models of NP

agglomeration

o
Basic approach

o
Monte Carlo numerical simulations

o
Comparison of predictions

with experimental data

o
Dependence of NP agglomeration

on basic system parameters

o
Future work


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0830117


eNMs

may be released to the
environment throughout their life
-
cycle


Preliminary
in vitro (with various cell
lines) and in
-
vivo
studies with simple
organisms (e.g.,
zebrafish
) suggest
that certain
eNMs

may be toxic at
certain exposure concentration levels


The transport and fate of
eNMs

in the
environment is governed by their
agglomeration state


The toxicity of
eNMs

may be impacted
by their primary size and their
agglomeration state


The removal of
eNMs

from aqueous
streams can be facilitated by
controlling their aggregation state


Motivation



Nanoparticle
Toxicity

Exposure

Fate &
Transport

Particle Size
Distribution

Particle
-
Cell
Interactions

Nanoparticle
Aggregation

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0830117

Environmental Multimedia
Fate & Transport
of eNMs

The transport and fate of nanoparticles is governed by their agglomeration state


Atmosphere
Water Body
Sediment
Soil
eNMs
input
Aerosolization
Sedimentation
Dry/wet Deposition
Resuspension
Flooding
Adsorption
Resuspension
Aggregation
Disaggregation
Adsorption
Desorption
Dispersion
Convection
Dry/wet Deposition
Runoff
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0830117

Environmental
Intermedia

Transport of Particles


Dry Deposition

Wind Soil Resuspension

Wet Scavenging

Aerosolization

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Atmospheric Deposition of Particles onto Water Surfaces


The dry deposition velocity of particles varies with particle size


Deposition Velocity (cm/s)

Particle Diameter, (µm)

1 nm

Diffusion

Impaction

1

10

10
-
2

10
-
3

10
-
1

1

10

10
2

0.1

0.01

Williams, R.M., A model for the dry deposition of particles to natural water surfaces.
Atmospheric Environment (1967), 1982. 16(8): p. 1933
-
1938

The dry deposition
velocity of particles
varies with particle size

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Rain Scavenging of Nanoparticles


Efficiency of NP removal from the atmosphere via wet
deposition depends on particle size
Cohen, Y. and P. A.
Ryan, "Multimedia Transport of Particle Bound Organics:
Benzo
(a)
Pyrene

Test Case,” Chemosphere, 15, 31
-
47 (1986).



Cohen, Y. and P. A. Ryan, "Multimedia Transport of Particle Bound Organics: Benzo(a)Pyrene Test
Case,” Chemosphere, 15, 31
-
47 (1986).

Efficiency of NP removal
from the atmosphere via
wet deposition depends on
particle size

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Gravitational Sedimentation of Nanoparticles in Aqueous Media

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eNM Size Distribution in Aqueous Systems


DLS is the standard approach to quantifying the size distribution
of nanoparticles


The reliability of DLS measurements is dependent on the NP
concentration and suspension stability


Suspension stability is impacted by NP agglomeration
(aggregation)/disaggregation which directly affect particle
gravitational sedimentation



90
°

~40μm

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Experimental Procedure


1000 ppm
suspension

Nanoparticle
powder

DLS

20ppm
suspension

Sonicate for 30
minutes in T
-
controlled bath

Sonicate for
5 minutes

Time delays between
consecutive steps ~5 s

Dilute

IS adjusted

pH adjusted
aqueous
solution

NPs: TiO
2

(21 nm, IEP=6.5, 21% A/79%R)


CeO
2

(15 nm, IEP=7.8)

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v
an der Waals Attraction





EDL Repulsion


2

cases, quantified

by inverse Debye Length
𝜅
(

)





















𝜅
𝑟
>
5

𝜅
𝑟
<
5

Φ
 𝐿
,

=
4𝜋𝜀
𝜀
𝑜
𝑟

𝑟

𝑌

𝑌

𝜓
𝑜
2
𝑇

2
exp
(

κH
)

+
𝑟

+
𝑟



Φ
 𝐿
,

=
4𝜋𝜀
𝜀
𝑜
𝜓
𝑜
2
𝑟

𝑟

𝑟

+
𝑟

ln
(
1
+
exp

𝜅
)

Φ
𝑣 𝑊
,

=

𝐴
𝐻
6
[
2
𝑟

𝑟

𝑅
2

𝑟

+
𝑟

2
+
2
𝑟

𝑟

𝑅
2

𝑟


𝑟

2
+
𝑛
𝑅
2

𝑟

+
𝑟

2
𝑅
2

𝑟

+
𝑟

2


]

eNP

eNP

Particle
-
Particle Interactions (Classical DLVO)

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DLVO Theory
(slide shows
forumlas

for types of interaction

Type of
Interactions

Expression

EDL

<5


EDL

>5

vdW


2 2
,
2 2 2 2 2 2
2 2 ( )
ln
6 ( ) ( ) ( )
i j i j i j
H
vdW ij
i j i j i j
rr rr R r r
A
R r r R r r R r r
 
 
    
 
     
 
 
r





2
,
4 ln 1 exp
i j
EDL ij o o
i j
rr
r r
 
    

r



2
,
exp
4
EDL ij o i j i j
i j
kT
rr YY
e H r r


 
 
 
 
 
 
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Particle
-
Particle Interactions


Classical DLVO only accounts for vdW and EDL


Classical DLVO assumes hard sphere


O.K. for environmental application as most frequently used
eNMs are spherical


Non
-
spherical particles exist


Nano
-
rod, nano
-
wire, etc.


DLVO does not account for:


Steric, hydration, magnetic, etc.


Modified DLVO can be utilized to account for additional
interaction energies and particle shape (e.g., sphericity)

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Size
distribution
of NPs in Aqueous Systems


Basis: Smoluchowski Coagulation Theory

𝜕
𝑛

𝜕𝑡
=
1
2

𝐾


𝑛


𝑛


=


1

=
1

=




𝑛


𝐾


𝑛



=
1
+

𝑛



𝑔

𝑝

,


𝑛


𝑔



=
1


𝐾


is the agglomeration frequency function:
𝐾

=
𝛽

𝑊






is the collision frequency:


=
2𝑇
3𝜇
𝑟

+
𝑟

1


+
1




𝑊


is the inverse sticking coefficient:
W
ij
=

2∙

exp

[
Φ
𝑇
,

𝑇
]

2
ds

2


Φ
𝑇
,


is the total interaction energy between


and



Estimated using classical DLVO theory


Time step to next agglomeration event:

2
ij
t
C N K
 
 
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0830117

Nanoparticle Brownian Motion & Settling


Stokes’ Settling
velocity




Diffusion length



2
2
x D t
   
6
B
k T
D
r



 


2
2
9
p f
sed
g r
v
 

 







r

<
x
>






=
𝑣


Δ
𝑡

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0830117

Monte Carlo Simulation of eNM
Agglomeration

.
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Constant
-
Number MC Simulations of Particles in a Box


Box is expanded to maintain
the particle concentration
upon aggregation events and
replenishment of particles to
maintain a constant number

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0830117

Simulations of Nanoparticles Agglomeration

Dynamic Monte Carlo
Simulation Solver

Primary NP Information

(e.g., primary size,
surface chemistry)

Solution Chemistry/Media
Parameters

(e.g., ionic strength, pH,
temperature, dielectric
constant)

Output:

-
Particle size distribution (PSD)

-
NP
concentration

Measured or Calculated Model
Parameters

(e.g.,
d
p
, zeta
potential,
IS
Hamaker
constant)

Aggregation
Model:

-

DLVO

-
Sedimentation

-
Particles in a “box”

Computational (Constant
-
Number Monte Carlo) model of NP agglomeration
making use of the DLVO theory accounting for NP sedimentation

Computational Cluster: 10 Nodes with a total of 20 Intel Quad
-
Core Xeon processors
(2.2


3.0 GHz) with 176 GB RAM

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Importance of Including Sedimentation in Model Simulations

Average
of 10 simulations of 5000
particles

CeO
2

TiO
2


ζ
CeO2

=
-
24.5 mV

ζ
TiO2

=
-
29 mV

A
H
,

= 42 zJ

A
H,

= 21 zJ

pH = 8,
IS
= 0.065 mM

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0830117

Convergence of Simulations

<d>
10

(nm)

Number of Simulation Particles

<d>
n

(nm)

Number of Simulations,
n

Expand box to
maintain mass
concentration
Determine diffusion
and settling distances
during the previously
determined time step
for all NPs
Final NP
Dispersion
Distribute
NPs in a box
End
Start
t
<
t
final
Calculate
agglomeration
frequency for all
NP pairs
Select a pair of NPs
for agglomeration
based on their
agglomeration
frequency
Calculate size and
position of
agglomerated pair
Calculate the time
step between the
agglomeration
events
Replenish particles
based on PSD
sampling to maintain
constant NP numbers
Replace particles
based on periodic
boundary condition
Replace particle
based on PSD of
settled particles
NPs diffuse
or settle out
of box
?
Yes
No
Diffuse
Settle
Neither
Number of Simulation Particles

Average of 10
simulations

S
mean particle size

(%)

S
mean particle size
,

nm

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Comparison of Experimental and Simulation Results

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eNP
(a)

Type

z

[mV]

(pH)

IS

[mM]

d
p
[nm]

d
exp

[nm]

d
sim

[nm]

%

abs
.

error
(b)

Jiang,

et

al
.


TiO
2

38

(
3
.
3
)

1

15

80

96

20
.
5

TiO
2

36

(
3
.
8
)

1

15

85

102

19
.
8

TiO
2

34

(
4
.
45
)

1

15

87

108

23
.
7

TiO
2

28
.
5

(
5
.
3
)

1

15

233

252

8
.
1

TiO
2

-
30

(
7
.
8
)

1

15

218

251

15
.
5

TiO
2

-
38

(
8
.
2
)

1

15

162

121

25
.
2

TiO
2

-
43

(
8
.
7
)

1

15

92

90

1
.
9

TiO
2

-
47
.
5

(
9
.
65
)

1

15

93

85

8
.
7

TiO
2

-
45

(
10
.
4
)

1

15

98

78

20
.
2

TiO
2

36

(
4
.
6
)

0
.
01

15

90

77

14
.
6

TiO
2

42

(
4
.
6
)

1

15

90

107

18
.
8

TiO
2

40

(
4
.
6
)

5

15

160

178

11
.
3

TiO
2

36

(
4
.
6
)

10

15

500

392

21
.
6

French,

et

al
.


TiO
2

35

(
4
.
5
)

4
.
5

5

90

109

20
.
8

TiO
2

35

(
4
.
5
)

8
.
5

5

500

632

13
.
5

TiO
2

35

(
4
.
5
)

12
.
5

5

700

628

10
.
3

Ji,

et

al
.


TiO
2

30
.
2

(
6
.
1
)

1

21

200

202

1
.
0

Present

Study

TiO
2

41

(
3
)

0
.
37

21

163

162

0
.
6

TiO
2

-
30

(
8
)

0
.
027

21

173

175

1
.
2

TiO
2

-
35

(
10
)

0
.
12

21

172

171

0
.
6

CeO
2

32

(
3
)

0
.
37

15

271

269

0
.
7

CeO
2

-
23
.
5

(
8
)

0
.
027

15

266

264

0
.
8

CeO
2

-
30

(
-
30
)

0
.
12

15

240

243

1
.
3

(b)

%

abs
.

error


(a) eNP


Engineered Nanoparticle

Summary of
Experimental
& Simulation
Conditions

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0830117

Particles Size Distributions (t=24 h)

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Dependence of TiO
2

Agglomeration on pH

Simulations:

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Dependence of Agglomerate Size on Ionic Strength

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Dependence of NP Agglomeration on the Hamaker Constant

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Dependence of Agglomerate Size on Primary NP Size

NP primary size



PSD tail
of small aggregates



Average NP aggregate size (in suspension)



For present primary size range:

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0830117

Summary and Future work

Monte Carlo (MC) simulations of NP agglomeration based on the Smoluchowski
equation and classical DLVO theory demonstrated reasonable quantitative
predictions of NP agglomeration (average size and size distribution) over a range of
solution conditions (pH= 3
-
10, IS= 0.03
-
12.5 mM for TiO
2

and CeO
2

NPs)

The present approach can be extended to include various modifications/extensions
of the DLVO theory

With extension and additional validation of the current modeling approach it will be
feasible to develop a practical parameterized model of NP agglomeration


New experimental DLS data are being generated over a wide range of conditions
specifically for extended model extension and validation


A machine learning approach is being developed to guide the task of data generation and
parameterized model development

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0830117

Questions?