The asymptotic Structure and Size Distribution of fractal-

like Aerosols made by Agglomeration

E. Goudeli, M.L. Eggersdorfer and S.E. Pratsinis

Particle Technology Laboratory, Institute of Process Engineering, Department of Mechanical and Process

Engineering, ETH Zürich, Sonneggstrasse 3, CH-8092 Zürich, Switzerland.

Agglomeration refers to the formation of physically attached primary particles by coagulation. It occurs in

environmental and industrial processes, especially in low temperature regions where sintering or

coalescence are rather slow. Understanding agglomeration is essential for optimal process design for

manufacture of nanomaterials as their fractal structure affects their handling and processing and

eventually their performance. The high particle concentrations encountered during nanomaterial

manufacturing lead to formation of agglomerates with well-defined asymptotic structure and size

distribution given by their fractal-like dimension and self-preserving size distribution, respectively.

The growth and detailed structure of fractal-like aerosol particles undergoing agglomeration is

investigated here from the free molecular to the continuum regime by discrete element modeling.

Particles in the free molecular regime follow ballistic trajectories described by an event driven method

whereas in the near continuum (gas-slip) and continuum regimes Langevin Dynamics describe their

diffusive motion. The simulations are validated by the attainment of the collision frequency and self-

preserving size distribution (SPSD) of fully coalescing particles in free molecular and continuum regimes

as well as the corresponding asymptotic fractal dimensions, Df, of 1.91 and 1.78 by ballistic and

diffusion-limited cluster-cluster agglomeration, respectively.

The evolution of agglomerate structure from perfect spheres (Df = 3) to the above well-known

asymptotic fractal-like structures is simulated in detail and a simplified expression is extracted that can be

readily used in process design for synthesis of nanomaterials or in environmental models for ambient

aerosols (e.g. air pollution and climate forcing). Fractal-like agglomerates exhibit considerably broader

SPSD than spherical particles when made by coagulation-agglomeration: the number-based geometric

standard deviation of the radius of gyration of agglomerates in the free molecular and continuum regimes

is 2.27 and 1.95, respectively, compared to that of spherical particles of 1.45. The quasi-self-preserving

geometric standard deviation of the radius of gyration of agglomerates exhibits a characteristic minimum

of 1.65 in the transition regime at Knudsen numbers, Kn ≈ 0.2. In contrast, their Df linearly shifts from

1.91 in the free molecular to 1.78 in the continuum regime.

Keywords: agglomeration, fractal dimension, self-preserving size distribution, discrete element method.

Coagulation - Agglomeration of

Fractal-like Particles: Structure &

Self-Preserving Size Distribution

E. Goudeli, M.L. Eggersdorfer and S.E. Pratsinis

Particle Technology Laboratory

ETH Zürich, Switzerland

2

Continuum

Mesoscale

Molecular

dynamics Quantum

mechanics

Buesser B, Pratsinis SE, Annual Rev. Chem. Biomol. Eng., 3 (2012) 103–127.

Multiscale Design of Aerosol Synthesis

3

Introduction

Structure: affects product characteristics (e.g. rheological

properties1) & performance, radiative forcing2, visibility3

1. Russel WB. (1987) Powder Technol., 51,15-25.

Reinforcing rubbers

Suspensions

(e.g. pigments)

Hard agglomerates:

catalysis, lightguide preforms, electroceramic devices

Soft agglomerates:

2. Martins JV, Artaxo P, Liousse C, Reid LS, Hobbs PV, Kaufman YJ. (1998) J. Geophys. Res., 103, 32041-32050.

3. Berry MV, Percival IC. (1986). Opt. Acta, 33, 577-591.

4

Introduction

Connect emissions to aerosol size measurements

Monitoring combustion emissions & atmospheric aerosols

IPCC, 2007

Negative

Radiative

Forcing

5

Motivation

Optimal aerosol reactor & process design

Close control of particle size & structure

catalytic activity and selectivity1

Open aggregates facilitate gas transport in and out of pellets

- Introduction of detailed structure (Df) in collision rates, β

,f cD d

Need a reliable Df descriptor

Controlled agglomeration can minimize costly separation

techniques

1. Christopher P, Linic S. (2010) Chem. Cat. Chem., 2, 78-83.

1. Eggersdorfer ML, Pratsinis SE. (2014) Adv. Powder Technol., 24, 71-90.

rm: mobility radius

free molecular regime

DMA

rva: volume-equivalent radius

rg: radius

of gyration

Light Scattering

Microscopy

Agglomerate Characterization

rm: mobility radius

continuum regime

2 10g

var

r

6

Filter weighing APM

TEOM

Previous Work

Brownian coagulation of spheres1,2

Non-spherical particles

determination of evolving structure4,5,6,7

4. Artelt C, Schmid H-J, Peukert W (2003), J. Aerosol Sci., 34, 511-534.

5. Kostoglou M, Konstandopoulos AG (2001), J. Aerosol Sci., 32, 1399-1420.

6. Schmid HJ, Al Zaitone B, Artelt C, Peukert W (2006), Chem. Eng. Sci., 61, 293-305.

7. Eggerdorfer ML, Kadau D, Herrmann HJ, Pratsinis SE (2011), Langmuir, 27, 6358-6367.

7

1. von Smoluchowski M. (1917), Z. Phys. Chem. Stoechiom. Verwandtschafts., 92, 129-168.

2. Buesser B, Heine MC, Pratsinis SE (2009), J. Aerosol Sci., 40, 89-100.

collisional growth and dynamics3

3. Rogak SN, Flagan RC (1992), J. Colloid Interface Sci., 151, 203-224.

8

SiO2 particles (like fumed silica), T = 27 oC

DLCA

2. Heine MC, Pratsinis SE. (2007). Langmuir, 23, 9882-9890.

Continuum regime Langevin Dynamics 2

Free molecular regime

Event-driven method 1

BCCA

3. Eggersdorfer ML, Kadau D, Herrmann HJ, Pratsinis SE. (2010). J. Colloid Interface Sci., 342, 261-268.

time

if N<N0/2

double

if N<N0/2

double

1. Allen MP, Tildesley DJ. (1991). Computer Simulation of Liquids, Oxford University Press, New York.

Simulation Method

9

10-3

10-2

10-1

100

101

102

0.3

1

5

β, 1

0-1

5, m

3/s

Kn = λair/rp

Free

molecular

(Kn >10)

Enhancement

due to

polydispersity

from the rapid

attainment of

SPSD

Validation – Full Coalescence

Collision Frequency Function, β

10-3

10-2

10-1

100

101

102

0.3

1

5

This work

1. Fuchs NA. (1964). Mechanics of Aerosols. Macmillan, New York. 2. Buesser B, Heine MC, Pratsinis SE. (2009). J. Aerosol Sci., 40, 89–100.

Fuchs1

7.34%

Continuum

(Kn <0.1)

17.9%

[2]

[2]

2 1

2 1

1 1

2N N

t t

10

Kn = λair/rp

3. Xiong Y, Pratsinis SE. (1991). J. Aerosol Sci., 22, 637-655. 4. Vemury S, Kusters KA, Pratsinis SE. (1994). J. Colloid Interf. Sci., 26, 175-185.

1. Otto E, Stratmann F, Fissan H, Vemury S. Pratsinis,SE. (1994). Part. Part. Syst. Char., 11, 359-366.

10-3

10-2

10-1

100

101

102

1.0

1.1

1.2

1.3

1.4

1.5

1.6

Xiong & Pratsinis

Otto et al. DEM simulations

Continuum

Vemury et al. Landgrebe

& Pratsinis

σg,v

σg,n = 1.44

σg,n = 1.46 σg,n

σg,v = 1.33 σg,v = 1.307

2. Landgrebe JD, Pratsinis SE (1989). Ind. Eng. Chem. Res., 28, 1474-1481.

σg

Validation – Full Coalescence

Geometric Standard Deviation

Free

molecular

Continuum

Free molecular

11

Agglomerate Dynamics - Size Evolution

Ave

rage

ag

glo

me

rate

ra

dii,

nm

1. Eggersdorfer ML, Pratsinis SE. (2014) Adv. Powder Technol., 24, 71-90.

2. Sorensen CM. (2011) Aerosol Sci. Technol., 45, 765-779.

Number of primary particles, np

33=27 times

overprediction of mass

1 10 100 100010

100

1000

1 10 100 100010

100

1000

1 10 100 100010

100

1000

rva

rg

rm np = 300 D

f ?

3a

g

v

r

r

Df = 1.91 – 1.78

10 100 10001

10

100

1000

10000

10 100 10001

10

100

1000

10000

10 100 10001

10

100

1000

10000

12

10 100 10001

10

100

1000

10000

10 100 10001

10

100

1000

10000

10 100 10001

10

100

1000

10000

10 100 10001

10

100

1000

10000

Num

be

r of pri

ma

ry p

art

icle

s,

np

rp,0 = 15 nm

Df = 3, kn =1

(t = 10-5s) Df = 1.71,

kn = 1.44

(t = 10-3s)

Df = 2.5, kn = 1.08

(t = 2∙10-4s)

Df = 1.76, kn = 1.34

(t = 10-1 s)

Agglomerate radius of gyration, rg, nm

Evolution of Agglomerate Structure by

Coagulation

fD

g

p n

p

rn k

r

1.05pn

2.4pn

616pn

19pn

10 100 10001

10

100

1000

10000

13

100

101

102

103

104

1.5

2.0

2.5

3.0

Number of primary particles, np

Evolution of Average Agglomerate Structure

PP radius:

1 nm

2 nm

7 nm

15 nm

100 nm

500 nm

( )f pD f n

Effe

ctive f

racta

l dim

ensio

n, D

f

1 10 100 1000 100001.5

2.0

2.5

3.0

14

Effective fra

cta

l dim

en

sio

n,

Df

1. Jullien R, Botet R. (1987). World Scientific, River Edge.

2. Tence M, Chevalier JP, Jullien R. (1986). J Phys, 47, 1989–1998.

KnD,m = λp/rm

10-3

10-1

101

103

105

1.5

2.0

2.5

3.0

10-3

10-1

101

103

105

1.5

2.0

2.5

3.0

Df = 1.91 ± 0.03

Free

molecular

(BCCA)

Continuum

(DLCA)

Df = 1.78 ± 0.05

Agglomerate Structure Evolution

,( )f D mD f Kn

10-3

10-1

101

103

105

1.5

2.0

2.5

3.0

15

Number of primary particles, np

Po

wer-

law

exponent, k

n

100

101

102

103

104

1.0

1.2

1.4

1.6

PP radius:

1 nm

2 nm

7 nm

15 nm

100 nm

500 nm

Evolution of Average Agglomerate Structure

fD

g

p n

p

rn k

r

kn = 1.4 ± 0.2

10-3

10-2

10-1

100

101

102

10-5

10-4

10-3

10-2

10-1

100

16

Agglomerate Self-Preserving Size Distribution

Norm

aliz

ed c

oncentr

ation, ηΨ

Dimensionless volume, η=Nυ/V

PBE, (Df = 1.78)3

DEM Continuum regime

PBE, (Df = 1.91)3

DEM

Free molecular regime PBE, (Df = 1.91)1,2

PBE, (Df = 1.78)1,2

Large particles

excellent agreement

in DEM and PBE

Small particles

DEM higher than

PBE due to

“interpenetration

effect”

more significant in

FM regime

1. Fuchs NA. (1964). Mechanics of Aerosols. Macmillan, New York.

3. Thajudeen et al. (2012). Aerosol Sci. Technol., 46, 1174–1186.

10-1

100

101

1020.1

1

10

β,

10

-15,

m3/s

Kn = λair/rp

Free molecular

Continuum Thajudeen et al.2

Fuchs1

Thajudeen et al.3

Fuchs1,2

Transition Thajudeen et al.3

Fuchs1,2

2. Mulholland, G.W.; Samson, R.J.; Mountain, R.D.; Ernst, M.H. Energy Fuels 1988, 2, 481-486.

10-3

10-2

10-1

100

101

102

103

1.0

1.5

2.0

2.5

10-3

10-2

10-1

100

101

102

103

1.0

1.5

2.0

2.5

KnD,m = λp/rm 1. Vemury S, Pratsinis SE. (1995). J. Aerosol Sci. 26: 175-185. 17

σg,g,FM

= 2.27

σg

σg,v σg,v,C

= 1.31

σg,v,FM

= 1.41

agglomerates

spheres

σg,g: rg-based

σg,v: rva-based

σg,m

σg,g

σg,v

σg,g

, ,( )g g D mf Kn

σg,g,C

= 1.95

10-3

10-2

10-1

100

101

102

103

1.0

1.5

2.0

2.5

10-3

10-2

10-1

100

101

102

103

1.0

1.5

2.0

2.5

10-3

10-2

10-1

100

101

102

103

1.0

1.5

2.0

2.5

10-3

10-2

10-1

100

101

102

103

1.0

1.5

2.0

2.5

Geometric Standard Deviation

(Df = 1.8) σg,v = 1.301 (Df = 2)

σg,v = 1.421

(Df = 1.91)

σg,g: rg-based

σg,m: rm-based

σg,v: rva-based σg,m,FM

= 2.03

σg,g, [2] σg,pa, [2]

2. Gröhn, A.J.; Eggersdorfer, M.L.; Pratsinis, S.E.; Wegner, K., (2014). J. Aerosol Sci. 73, 1-13.

18

Conclusions

• Geometric Standard Deviation:

, ,( )g g D mf Kn

• Df evolution from spherical to

fractal-like particles: ( )f pD f n

,( )f D mD f Kn• Structure:

Transition regime:

linear function

19

THANK YOU FOR YOUR

ATTENTION!