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 Cooperative 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.

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OUTLINE

Motivation Toward predictive models of NP

agglomerationo Basic approacho Monte Carlo numerical simulationso Comparison of predictions

with experimental datao Dependence of NP agglomeration

on basic system parameterso Future work

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• 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

ExposureFate & Transport

Particle Size Distribution

Particle-Cell Interactions

Nanoparticle Aggregation

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Environmental Multimedia Fate & Transport of eNMs

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

Atmosphere

Water Body

SedimentSoil

eNMs input

Aerosolization

Sedimentation

Dry/wet Deposition

Resuspension

FloodingAdsorption

Resuspension

Aggregation

DisaggregationAdsorption

Desorption

Dispersion Convection

Dry/wet Deposition

Runoff

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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

Dep

ositi

on V

eloc

ity (

cm/s

)

Particle Diameter, (µm)

1 nm

Diffusion

Impaction

1 10

10-2

10-3

10-1

1

10

102

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

Detector

90°~40μm

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1000 ppm suspension

Nanoparticle powder

Detector

DLS20ppm

suspension

Sonicate for 30 minutes in T-

controlled bath

Sonicate for 5 minutes

Time delays between consecutive steps ~5 s

Dilute

IS adjustedpH adjusted

aqueous solution

NPs: TiO2 (21 nm, IEP=6.5, 21% A/79%R) CeO2 (15 nm, IEP=7.8)

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• van der Waals Attraction

• EDL Repulsion– 2 cases, quantified

by inverse Debye Length

𝜅𝑟>5

𝜅𝑟<5 Φ𝐸𝐷𝐿 ,𝑖𝑗=4𝜋𝜀𝜀𝑜𝑟 𝑖𝑟 𝑗𝑌 𝑖𝑌 𝑗𝜓𝑜2 (𝑘𝑇𝑒 )

2 exp (−κH )𝐻+𝑟 𝑖+𝑟 𝑗

Φ𝐸𝐷𝐿 ,𝑖𝑗=4𝜋𝜀𝜀𝑜𝜓𝑜2 𝑟 𝑖𝑟 𝑗

𝑟 𝑖+𝑟 𝑗

ln(1+exp (−𝜅𝐻 ))

Φ𝑣𝑑𝑊 ,𝑖𝑗=−𝐴𝐻

6[

2𝑟 𝑖𝑟 𝑗

𝑅2− (𝑟 𝑖+𝑟 𝑗 )2+

2𝑟 𝑖𝑟 𝑗

𝑅2− (𝑟 𝑖−𝑟 𝑗 )2+𝑙𝑛

𝑅2− (𝑟 𝑖+𝑟 𝑗 )2

𝑅2− (𝑟 𝑖+𝑟 𝑗 )2 ]

eNP eNP

Particle-Particle Interactions (Classical DLVO)

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DLVO Theory (slide shows forumlas for types of interactionType of Interactions Expression

EDL<5

EDL>5

vdW2 2

, 2 2 2 2 2 2

2 2 ( )ln

6 ( ) ( ) ( )i j i j i jH

vdW iji j i j i j

rr rr R r rA

R r r R r r R r r

r

2, 4 ln 1 expi j

EDL ij o oi j

rr

r r

r

2

,

exp4EDL ij o i j i j

i j

kTrr 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

– is the agglomeration frequency function: – is the collision frequency: – is the inverse sticking coefficient:– is the total interaction energy between and

• Estimated using classical DLVO theory

– Time step to next agglomeration event:

2

ij

tC N K

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Nanoparticle Brownian Motion & Settling

• Stokes’ Settling velocity

• Diffusion length22x D t

6Bk T

Dr

22

9p f

sed

g rv

r <x>

𝑑𝑠𝑒𝑑

𝑑𝑠𝑒𝑑=𝑣𝑠𝑒𝑑 ∙ Δ 𝑡

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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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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., dp, 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

CeO2TiO2

ζCeO2 = -24.5 mV ζTiO2 = -29 mV AH, = 42 zJAH, = 21 zJ

pH = 8, IS= 0.065 mM

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Convergence of Simulations

<d>

10 (

nm)

Number of Simulation Particles

<d>

n (n

m)

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<tfinal

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

Sm

ean

part

icle

siz

e (%

)

Sm

ean

part

icle

siz

e, n

m

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Comparison of Experimental and Simulation Results

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eNP(a) Type

z [mV] (pH) IS [mM] dp[nm] dexp [nm] dsim [nm] % abs. error(b)

Jiang, et al. TiO2 38 (3.3) 1 15 80 96 20.5TiO2 36 (3.8) 1 15 85 102 19.8TiO2 34 (4.45) 1 15 87 108 23.7TiO2 28.5 (5.3) 1 15 233 252 8.1TiO2 -30 (7.8) 1 15 218 251 15.5TiO2 -38 (8.2) 1 15 162 121 25.2TiO2 -43 (8.7) 1 15 92 90 1.9TiO2 -47.5 (9.65) 1 15 93 85 8.7TiO2 -45 (10.4) 1 15 98 78 20.2TiO2 36 (4.6) 0.01 15 90 77 14.6TiO2 42 (4.6) 1 15 90 107 18.8TiO2 40 (4.6) 5 15 160 178 11.3TiO2 36 (4.6) 10 15 500 392 21.6

French, et al. TiO2 35 (4.5) 4.5 5 90 109 20.8TiO2 35 (4.5) 8.5 5 500 632 13.5TiO2 35 (4.5) 12.5 5 700 628 10.3

Ji, et al. TiO2 30.2 (6.1) 1 21 200 202 1.0Present Study TiO2 41 (3) 0.37 21 163 162 0.6

TiO2 -30 (8) 0.027 21 173 175 1.2TiO2 -35 (10) 0.12 21 172 171 0.6CeO2 32 (3) 0.37 15 271 269 0.7CeO2 -23.5 (8) 0.027 15 266 264 0.8CeO2 -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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Particles Size Distributions (t=24 h)

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Dependence of TiO2 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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Summary and Future workMonte 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 TiO2 and CeO2 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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Questions?