CHARACTERISTICS OF SOOT AGGREGATION INCONVECTIVE ENVIRONMENTS

G. Inci1, A. Kronenburg1, A. Arnold2

1 Institut fur Technische Verbrennung, Universitat Stuttgart,Herdweg 51, 70174 Stuttgart, Germany

2 Institut fur Computerphysik, Universitat Stuttgart,Allmandring 3, 70569 Stuttgart, Germany

kronenburg@itv.uni-stuttgart.de

AbstractThe sensitivity of the dynamics of soot aggrega-

tion and cluster growth on the surrounding flow condi-tions has been studied here. Langevin dynamics (LD)are used to track particle trajectories and collisions in“quasi” turbulent flows with different flow and particleconditions that can be characterized by the Peclet andKnudsen numbers. The analysis shows a clear depen-dence of the aggregates’ morphologies on the aggre-gates’ sizes and on their growth histories, and inclu-sion of these dependencies will be pivotal for accuratestatistical modelling of the collision frequencies in thecontext of population balance modelling.

1 IntroductionSoot aggregates are emitted from many combus-

tion processes as a result of incomplete combustionof a hydrocarbon. The hydrocarbon molecules nu-cleate through a complex series of chemical reactionsand grow via surface addition mechanisms, e.g. sur-face reactions and condensation [Frenklach (2002)].The newly formed particles then collide due to theirrelative velocity that is induced by Brownian motionand/or by a surrounding fluid flow. The particles maystick to each other due to covalent bonding, and sootaggregates are formed [Tsantilis and Pratsinis (2004)].The aggregates are found to be of very different sizesand shapes and their irregularity has a significant im-pact on their – usually adverse – environmental andhealth effects. These health issues have motivated con-siderable interest in aggregation kinetics since the ag-gregation process (via the collision and adhesion) pri-marily characterizes the morphology of soot [Maricq(2007)].

Many attempts have been made to model the aggre-gation process of soot particles in quiescent fluids andas a result, the structure of such aggregates is known insuch cases [Megaridis and Dobbins (1990)]. However,during the combustion processes, the surrounding fluidis not quiescent and the fluid flow significantly affectsthe aggregate kinetics of growing soot and hence sootmorphology [Selomulya (2001)]. Therefore, new ap-proaches require a comprehensive aggregation model

for soot nanoparticles that account for well-definedinteractions between the particles and hydrodynamicconditions that the particles are placed in.

Recent experimental [Wengeler and Nirschl (2006)]and numerical studies [Schlauch et al. (2010); Zei-dan et al. (2007)] have investigated the behaviour ofsoot and/or colloidal aggregates within simple shearflow. These studies mostly highlight the aggregatefragmentation due to shear. For fragmentation to oc-cur, the hydrodynamic stresses along aggregate tracksmust surpass the bond strength [Wengeler and Nirschl(2006)]. Lahuij et al. (2013) studied the behaviour ofsoot aggregates under stress and showed that the ag-gregates resist to break-up, and indeed, the conditionsstudied by Wengeler and Nirschl (2006) are somewhatatypical for a combustion process or any nanoparticleflame synthesis. Similarly, Rothenbacher et al. (2008)showed that high impact velocities (ufluid > 40 m/s)are required to induce fragmentation of the aggregates.We will therefore focus here on conditions that aremore typical for a combustion process and consideraggregation only.

The statistics of aggregation are typically describedby aggregation source and sink terms in a populationbalance equation,

∂N(t, x)

∂t=

1

2

∫ x

0

β(t, x, x− x)N(t, x)N(t, x− x)dx

−∫ ∞0

β(t, x, x)N(t, x)N(t, x)dx. (1)

The key to the accurate modelling of aggregation isthe right choice of the collision frequency, β, betweenaggegates of different sizes. Meyer & Deglon’s com-prehensive review [Meyer and Deglon (2011)] dis-cusses all standard expressions for β and many mod-els were validated by numerous direct numerical sim-ulation studies (e.g. [Mei and Hu (1999); Sundaramand Collins (1997)]), but only very few studies haveconsidered non-spherical particles. However, signif-icant effects of the aggregates’ structure, usually de-fined by the fractal dimension Df , on the aggregationdynamics can be expected [Friedlander (2000)], andcollision frequencies are commonly modified by in-clusion of Df as parameter describing the aggregates’shape. Common expressions for the collision frequen-cies, β(Rg,i, Rg,j) of agglomerates of size (radius of

gyration) Rg,i and Rg,j are, e.g.

β =2kbT

3µ

(R

−3Df

g,i +R−3Df

g,i

)(R

3Df

g,i +R3

Df

g,i

)2

(2)

β = 1.3( εν

)1/2 (R

3/Df

g,i +R3/Df

g,i

)3(3)

for Brownian particles and for particles in turbulentflows, respectively. Usually, a constant Df is assumedfor the entire particle range and simulation time.

The fractal dimension, however, changes during thecourse of aggregation and is dependent on the (turbu-lent) flow field and the relative importance of Brown-ian motion [Meakin (1983)]. The current paper there-fore investigates the dynamics of the aggregation pro-cess and the resulting morphology of the aggregatesin environments of varying characteristic conditions.We use Langevin dynamics (LD) to model the aggre-gation of primary particles and the setup shall mimicaggregation of soot particles in the post flame regionof combustion processes.

The next section of this paper presents the numericalapproach and introduces the characteristic quantitiesfor the aggregation in turbulent advective enviroments.The third section analyses the resulting agglomerates’morphology in terms of these characteristic quantities.

2 Numerical method

We use the open source molecular dynamics (MD)software package ESPRESSO to model the aggrega-tion of small particles [Arnold et al. (2013)].

Interparticle interactionsThe non-bonded potential between two primary par-

ticles is modelled by a standard Lennard-Jones-(12,6)-potential. Integration over all molecules within the re-spective particles yields an effective potential betweentwo spherical particles of diameter σ. The integratedpotential can be written as [Lazaridis and Drossinos(1998); Narsimhan and Ruckenstein (2008)]

U(r) = −A6

(ln

(r2 − σ2

r2

)+

σ2

2(r2 − σ2)+

σ2

2r2

),

(4)where r is the distance between the two respective par-ticles, σLJ is the distance at which the potential is zeroand A is the Hamaker constant and expressed by

A = 4πεσ6LJn

2. (5)

The parameter n is the molecular number density inthe solid [Isella and Drossinos (2010)]. The Hamakerconstant is set to A = 2.38 ∗ 10−19J which is a typ-ical value for hydrocarbon primary particles [Rothen-bacher et al. (2008)]. Also, ε is set to 1.5% of the par-ticle diameter and a cut-off distance of rcut = 5σ hasbeen introduce for numerical efficiency. Beyond thisdistance the attraction between particles is negligible

compared to the thermal energy and particle interac-tions can be neglected.

Particle aggregation modelClassical MD simulations with spherically symmet-

ric potentials model Brownian motion and subsequentaggregation quite accurately. However, rolling andsliding of the primary particles after collision leadsto rather compact (and unrealistic) agglomerates. Amore realistic resulting aggregate structure can be en-sured by following the approach outlined in Inci et al.(2014). Two colliding particles are assumed to stickpermanently at the point of first contact. Contact is de-fined when their center-to-center distance, r, reducesto the particle diameter σ + ε with ε being small. Thecontacting particles are bound via a 2-body spring po-tential and 3-body angular potential. The 2-body har-monic bond potential exerts adhesion force betweenthe centers of the contacting particles. Its distance-dependent energy configuration is given by

Uharmonic−bond =1

2kh(rij − r0)2, (6)

where rij = ‖~rj − ~ri‖ gives the distance betweenthe particles, r0 is the equilibrium distance and kh isthe bonding constant. The constant kh can be relatedto the adhesion energy between two particles by theJKR theory

(kh = 2EJKRadh /(rij − r0)2

)with [Chok-

shi et al. (1913)]

EJKRadh = Ψπa2. (7)

The parameter a is the radius of the circular contactregion and is related to the Young’s modulus, EY , tothe Poisson’s ratio, υ, and to the surface energy Ψ by

a =

(3Ψπ(σ/2)2

2K

)1/3

,

K =2EY

3(1− υ2),

Ψ ≈ A

24πD20

. (8)

D0 can be set to D0 = 0.165 nm [Pantina andFurst (2006)] and the Hamaker constantA is estimatedfrom experimentally measured fragmentation curvesfor diesel soot [Rothenbacher et al. (2008); Chokshiet al. (1913)].

In the (standard) case of more than two particleswithin the aggregate, an additional angular poten-tial fixes the angle between the position vectors fromthe center particle to adjacent particles. This proce-dure prevents the particles from sliding and/or rollingaround their contact point. The energy configurationis given by

Uangle = ka(θ − θ0)2, (9)

where θ is the angle in radians between the vectors~rij = ~rj − ~ri and ~rkj = ~rj − ~rk, θ0 is the equilibrium

angle, and ka is the angle bond constant. The anglebond constant, ka is estimated by [Becker and Briesen(2008)]

ka =κ0192

, (10)

with the single-bond bedinding rigidity κ0 [Pantinaand Furst (2006)]

κ0 =3πa4EY4(σ/2)3

. (11)

Equations of motionThe random motion of the interacting particles in

a quiescent flow is modelled by solving the Langevinequations of motion. The movement of the particlesis affected by thermal kinetics and viscous drag. TheLangevin equation for the i-th particle [Kubo (1966)]is then given by

mv = FC − γm(vp − vf ) + W, (12)

where FC are conservative forces, γ is the friction con-stant between the primary particle and the surround-ing fluid, vp and vf are the particle and fluid veloci-ties, respectively, and Wi(t) is Einstein’s white noiseterm. The latter models the random kicks of the fluidmolecules onto the particles. The velocity differencein equation (12) ensures the hydrodynamic drag on aparticle being proportional to the relative velocity ofthe particle with respect to the surrounding fluid.

Characteristic quantitiesWe attempt to simulate aggregation of particles in

turbulent environments. Characteristic time scales de-scribing the motion of particles are then given by thetime scales for the fluid flow (τfl), the Brownian mo-tion (τBM ) and the particle relaxation (τp). The rele-vant time scale for a turbulent flow is the Kolmogorovtime scale,

τfl = τK =(νε

)1/2, (13)

with ε denoting the dissipation rate and ν denoting thekinematic viscosity of the fluid. The time scale char-acterizing the Brownian diffusion of particles (τBM )represents the time taken by a particle to diffuse a dis-tance equal to its radius and can be approximated byr2p/D where rp is the particle radius andD is the diffu-sion coefficient. The Brownian diffusion time is thengiven by

τBM =σ2

4D=

σ2

4 kBT3πµσ

=3

4

πµσ3

kBT. (14)

Finally, the particle relaxation time is defined by

τp =ρpσ

2

18µ. (15)

The relevant time scales are now given by the dimen-sionless Stokes, Peclet and Knudsen numbers. The

Stokes number is defined as

St =τpτfl

=ρpσ

2

18µ

( εν

)1/2, (16)

it is much smaller than unity for the particles underconsideration here, and a dependence of the aggrega-tion process on St is expected to be negligible for thepresent study. This is in contrast to the Peclet number,defined as

Pe =τBMτfl

(17)

that characterizes the relative importance of trans-port by convection and transport by Brownian mo-tion. It also characterizes the relative importance of therespective mechanisms inducing relative motion be-tween the particles and leading to collision [Mokhtari(1998)]. Lastly, the Knudsen number which is givenby Kn = 2l/σ with l being the mean free path, is in-dicative of the magnitude of Brownian motion of theparticles.

Characterization of the carrier flow fieldThe turbulent flow field shall be represented by pairs

of counter-rotating vortices which are periodic in 2spatial directions and invariant in the third direction asillustrated in Fig. 1. These very simplified represen-

Figure 1: Representative vortex pair with relatively constantshear throughout the domain.

tations of a (turbulent) flow field seem to be adequatefor two reasons: (1) The typical size of primary sootparticles is much smaller than the smallest turbulencescale and aggregates close in space will experience arelative constant shear while approaching and collid-ing. Here, the counter-rotating vortices represent theKolmogorov eddies and are at least two orders of mag-nitude larger than the particle diameter. Larger eddieswill not directly influence the aggregation process; (2)it allows for a better (unambiguous) characterizationof the flow conditions and relevant time scales due toa relatively uniform dissipation rate (i.e. absence ofintermittency) and numerically, a simple coupling be-tween the particle motion and the flow field can be re-alized.

The streamlines of a circular flow are concentric cir-cles with their streamline functions given by

ψ =

{12c1r

2 , if 0 6 r 6 L/4

c2r∗2sin2θ , otherwise.

(18)

The first represents the stream function of a vortex,with a vortex strength of c1 and the vortex radius of r,the second represents the stream function of a cornerflow, in which c2 represents the vortex strength, andr∗ is the vortex radius. The superscript star indicatesa different coordinate system with its origin situatedin the corner. Matching conditions at the edge of thevortex give

c2 (x, y) =c12× (19)2R ln y +√R2 − y2 −R ln

(R+

√R2 − y2

)+ cx

R− y

−2R lnx+

√R2 − x2 −R ln

(R+√R2 − x2

)+ cy

R− x

}

with R being the vortex radius and x and y represent-ing cartesian coordinates within a coordinate systemcentered at an individual vortex origin. The counterro-tating vortexes will lead to accumulation of aggregatesin regions of low vorticity and will influence the rateof aggregation and the aggregates morphology. Wetherefore superpose eddies of different sizes. A trans-lational shift of the superposed fields by a quarter ofthe vortex sizes avoids an overlay of all low vortic-ity regions. The maximum eddy size of 2R = L/2is limited by the box length L. The smallest eddysize will be given by the Kolmogorov lenght scale

lK =(ν3

ε

)1/4. Here, the computational domain spans

8 Kolmogorov length scales and eddies of three differ-ent sizes can be superposed. Figure 2 shows the ve-locity fields of the smallest and the largest eddies. The

Figure 2: Velocity fields for the different eddy sizes.

vortex strengths shall mimic the high frequency (smallscale) range of the turbulence spectrum. The energyratio of the different eddies is given by

E(12k)

E(k)=

(k2

)−5/3

k−5/3=

(1

2

)−5/3

(20)

This leads to c1,k/2 = 0.89c1,k. c1 of the smallestvortex is given by the desired dissipation rate

ε = 2νSijSij with Sij =1

2(ui,j + uj,i) .(21)

and the Kolmogorov scales.

An identical vortices arrangement is then introducedin the x-z- and y-z-planes as illustrated in Fig. 3 for aset of 4 vortices and the final velocity field is illus-trated in Fig. 4. The distribution of the dissipation ofthis velocity field is rather narrow (the local dissipa-tion rates of more than 90 % of the entire domain liewithin 1 order of magnitude) and the effect of shear onaggregation dynamics is thus quantifiable more easilythan in a real turbulent flow field.

Figure 3: Velocity magnitude for 3D arrangement of eddies.

Figure 4: Velocity magnitude of final superposed 3D eddies.

3 Results and Discussion

The primary objective of this paper is the analysisof the dependence of the agglomerates’ structures onthe surrounding conditions. Here, it had be hypothe-sized that the relative importance of Brownian motionand (turbulent) shear and their interplay have a largeinfluence on the dynamics of the aggregation pro-cess. Characteristic time scales that help to parame-terize these effects have been identified in the previoussection and table 1 lists the different simulations thataid the analysis. The general setup simulates the ag-glomeration of soot nanoparticles (ρ = 1500kg/m3)in air at a temperature of T = 600K. The meanfree path and the viscosity of air are then given bylmfp = 115nm and ν = 51 · 10−6m2/s.

Agglomeration of primary particle with three dif-ferent diameters (σ = 5 nm, σ = 20 nm, and

Table 1: Dimensional and characteristic quantities for the different cases

c-1 c-2 c-3 c-4 c-5 c-6 c-7 c-8 c-9 c-10 c-11 c-12dimensional characteristics

σ [nm] 5 20 100ε [m2/s3] 0 4.8·109 4.8·1011 4.8·1013 0 1.2·106 1.2·108 1.2·1010 0 7.5·101 7.5·103 7.5·105

time and length scalesτBM [s] 1.07·10−9 6.87·10−8 8.59·10−6

τp [s] 6.9·10−11 1.1·10−9 2.76·10−8

τK [s] 0 1.1·10−7 1.1·10−8 1.1·10−9 0 6.8·10−6 6.8·10−7 6.8·10−8 0 8.5·10−4 8.5·10−5 8.5·10−6

lK [m] 0 2.4·10−6 7.6·10−7 2.4·10−7 0 1.9·10−5 6.1·10−6 1.9·10−6 0 2.1·10−4 6.8·10−5 2.1·10−5

characteristic quantitiesKn [-] 45.98 11.49 2.29Pe [-] 0 0.01 0.1 1.0 0 0.01 0.1 1.0 0 0.01 0.1 1.0St [-] 0 6.5·10−4 6.5·10−3 6.5·10−2 0 1.6·10−4 1.6·10−3 1.6·10−2 0 3.2·10−5 3.2·10−4 3.2·10−3

σ = 100 nm) in three different advective environ-ments are investigated. The corresponding character-istic quantities are the Knudsen numbers of Kn = 46,Kn = 11.5, Kn = 2.3, and Pe = 0.01, Pe = 0.1,Pe = 1, respectively. Peclect numbers equal or be-low one were chosen since agglomerates grow andlarger Pe will arise during the simulations. In ad-dition, simulations with Pe = 0 (i.e. agglomera-tion due to Brownian motion only) serve as refer-ence cases. Every computation comprises 100.000particles, 5 runs have been conducted for all cases toachieve improved statistics in particular of larger ag-gregates. Note that the maximum dissipation rate israther high, typical values in jet flames are of the orderof ε = O(105)− O(106)m2/s3, but may be achievedin rather turbulent channel flows.

Results from one reference case with Pe = 0(case c-5) are presented in Fig. 5 and show very goodqualitative and quantitative agreement with experi-ments. The computed fractal dimension of the aggre-gate equals Df = 1.8 which is very close to typicalvalues for soot aggregates.

Figure 5: A comparison of aggregates (a) from MD and, (b)from TEM imaging. [Tumolva et al. (2010)].

The fractal dimension is computed from the slopeof the logarithm of the radius of gyration, Rg , ver-

sus the number of particles within the agglomerate, N[Xiong and Friedlander (2001)] as illustrated in Fig. 6.A moderate dependence of the gradient on the Knud-sen number can be observed. Larger Kn tend to yieldlower gradients and the evolution of the particles’ mor-phology warrants further investigation.

1

10

100 1000

log

Rg

log N

c1, Df=1.97

c5, Df=1.77

c9, Df=1.61

Figure 6: Radius of gyration, Rg , plotted versus the num-ber of particles in the aggregates, N, for differentKnudsen numbers with Pe = 0 (cases c-1, c-5 andc-9 from table 1).

In addition, Figure 7 demonstrates that the fractaldimension evolves with time and the average morphol-ogy of the aggregates tends to increase but with dif-ferent rates that depend on the relative importance ofthe flow field and the Brownian motion. For bettercomparison, the time has been normalized by the Kol-mogorov time scale, i.e. t∗ ≡ t/τK . This normaliza-tion assumes the turbulent shear being the major driverfor the relative motion between the different agglom-erates leading to collision, and this is certainly true forthe higher Pe number cases.

It is apparent from fig. 7 that the dynamics of theaggregation process are strongly dependent on Kn andPe and lead to differences in the average morphologyof the aggregates. This is also visualized in Fig. 8for t∗ = 6.4. Here Pe = 0 is approximated byPe = 10−5 to allow for a logarithmic presentation ofthe dependence of Df on Pe. Large Kn tend to formmore compact agglomerates due to the increased im-

Figure 7: Evolution of the fractal dimension with time fordifferent Kn and Re.

portance of Brownian motion while collision due tostrain tends to lead to more elongated agglomerateswith lower Df due to the preferential alignment oflarge clusters with the velocity gradients. This effectis somewhat reduced at larger Pe due to the reducedrelative importance of Brownian motion with respectto the collision induced by shear. Most importantly,these –rather plausible– observations are contradic-tory to early (common) presumptions by e.g. Meakin(1983).

5 10 15 20 25 30 35 40 45 1e-06 1e-05

0.0001 0.001

0.01 0.1

1

1.5

1.6

1.7

1.8

1.9

2

avg Df

Kn

Pe

avg Df

1.6

1.65

1.7

1.75

1.8

1.85

1.9

Figure 8: Fractal dimension as function of Kn and Pe at t∗ =6.4.

It is noted that the Pe dependence reduces duringthe course of the simulation and that –while differ-ences of ∆Df = 0.2 can change the collision ratesby a factor of 2– the overall effect of the different flowfields does not seem to be too significant. This can beexplained by the growth of the agglomerates and thestrongly non-linear dependence of Pe on characteris-tic particles size (σ, or –for larger agglomerates– theradius of gyration). This leads to rather large Pe andsmall Kn once the larger agglomerates have formed,the characterization of the dynamics by the initial flowconditions (initial Pe and Kn) may be misleading for asignificant fraction of the primary particles, since the

effect of the Brownian motion on large agglomerateswill disappear.

Averaged quantities are therefore not suitable forcharacterizing the dynamics of the aggregation processand a conditioning on the agglomerate’s size is advis-able. Figure 9 (top) shows the agglomerate size distri-butions at t∗ ≈ 10 for constant Pe and varying Kn.Dependencies are noticeable even though the timeshave been normalized by τK (which is equivalent inthis case to a normalization by τBM ). As expected,large Kn lead to much larger agglomerates with a si-multaneous reduction of small clusters. Similarly, anincrease in Pe reduces the number of small clusters andenhances cluster growth (cf. Fig. 9, bottom).

Figure 9: Agglomerates’ size distributions for different Kn(top, for Pe=0.1, t∗ ≈ 10) and for different Pe(bottom, Kn=11, t∗ ≈ 6.4).

More important, however, is the characterization ofthe morphology due to its pivotal role in determiningthe collision frequencies and thus the agglomerationdynamics (cf. eqs. (2) and (3)). Here, strong depen-dencies of Df on the agglomerate’s size can be ob-served. These trends are rather independent of the flowand primary particle characteristics for the smaller ag-glomerates as can be seen in Fig. 10. Due to the largenumber of smaller agglomerates (N < 300 particles,Rg < 5) the averaged characteristics are dominatedby these clusters, but do not represent the agglomer-ates’ shapes for the larger clusters well. The statisticsfor larger agglomerates are a bit noisy due to a lower

Figure 10: Dependence of the fractal dimension as functionof the agglomerate’s size (characterised by the ra-dius of gyration, Rg , for different Pe (Kn=11)(top) and Kn (Pe=0.1) (bottom) at t∗ ≈ 6.4.

sample size. However, they seem to indicate a level-ling off at Rg > 10 and approximate values at aroundDf = 2 with a maximum of Df = 2.3 which cor-responds to the average fractal dimension of the mostcompact aggregates in case of diffusion limited aggre-gation (DLA). Independent of the apparent noise, cleardifferences can be observed for the large clusters’ mor-phologies. Figure 11 shows 2 representative agglomer-ates for different Pe with the – for the cluster size – av-erage fractal dimension. The first agglomerate is muchmore compact than the agglomerate formed for Pe=1.This is due to the rather low influence of Brownianmotion even at the start of the simulation. Small clus-ters quickly align with the shear and only experiencevery little growth in the direction of the velocity gradi-ents (i.e. perpendicular to their alignment). The sim-ulated fractal dimension of around Df = 1.8 agreeswell with data from Megaridis and Dobbins (1990) forpure cluster-cluster collisions.

4 ConclusionsWe have analyzed the dynamics and the morphol-

ogy of soot aggregation using Langevin dynamics sim-ulations. Clear differences can be observed in the dy-namics of the aggregation process resulting in differ-

Figure 11: Representative agglomerates from case c-6 (top)and c-8 (bottom) with Rg ≈ 9 at t∗ = 6.4. Thefractal dimension are Df = 2 and Df = 1.84,the cluster sizes are N = 577 and N = 572particles, respectively.

ences in the evolution of the agglomerate’ size distri-bution even if a suitable time normalization is intro-duced. The dependence of the average structure ofthe agglomerate on Pe and Kn is, however, not thatpronounced. In contrast, a strong and lasting depen-dence of the morphology on the agglomerate’s sizecan be observed. In addition, larger agglomerates aremuch more susceptible to difference in flow and par-ticle characteristics. This is due to the agglomeratesgrowth history: large Kn and low Pe enhance aggrega-tion due to Brownian motion during the early stages ofthe growth process and provide more compact buildingblocks for the larger clusters formed at the later stagesof the process. Modelling Df (Rg,Kn, Pe) has notbeen attempted here, but further complementary stud-ies will now aim at finding improved dependencies ofthe collision kernels on Df (e.g. eq. (3)) within thecontext of population balance modelling. In addition,the present database will be used to assess existingmodels for β for agglomeration processes subject toBrownian motion and/or shear.

Acknowledgments

The authors acknowledge the support by DFG aspart of the Collaborative Research Center (SFB) 716.We would like to thank R. Weeber for his support withthe LD simulations.

References

Arnold, A., O. Lenz, S. Kesselheim, and et. al. (2013).Espresso 3.1: Molecular dynamics software forcoarse-grained models. In Meshfree Methods forPartial Differential Equations VI, Volume 89, pp.1–23. Springer Berlin Heidelberg.

Becker, V. and H. Briesen (2008, Dec). Tangential-force model for interactions between bonded col-loidal particles. Phys. Rev. E 78, 061404.

Chokshi, A., A. G. G. M. Tielens, and D. Hollenbach(1913). Dust coagulation. The Astrophysical Jour-nal 407, 806–819.

Frenklach, M. (2002). Reaction mechanism of sootformation in flames. Physical Chemistry ChemicalPhysics 4, 2028–2037.

Friedlander, S. K. (2000). Smoke, Dust and Haze (sec-ond ed.). Oxford University Press, Oxford, NewYork. ISBN 0-19-512999-7.

Inci, G., A. Arnold, A. Kronenburg, and R. Weeber(2014). Modelling nano-particle agglomeration us-ing local interactions. Aerosol Science and Technol-ogy DOI:10.1080/02786826.2014.932942.

Isella, L. and Y. Drossinos (2010). Nanoparticle ag-glomeration by langevin simulations. Phys. Rev.E 82, 011404.

Kubo, R. (1966). The fluctuation-dissipation theoremand Brownian motion. New York: Summer Lecturesin Theoretical Physics.

Lahuij, I., F. Dassenoy, B. Vacher, K. S. nadD. A. Brass, and M. Devine (2013). Understand-ing the deformation of soot particles/agglomeratesin a dynamic contact: Tem in situ compression andshear experiments. Springer.

Lazaridis, M. and Y. Drossinos (1998). Multilayer re-suspension of small identical particles by turbulent-flow. Aerosol Sci. Technol. 28, 548–560.

Maricq, M. M. (2007). Coagulation dynamics offractal-like soot aggregates. J. Aerosol Sci 38, 141–156.

Meakin, P. (1983). Effects of cluster trajectories oncluster-cluster aggregation: A comparison of lin-ear and brownian trajectories in two- and three-dimensional simulations. Physical Review A 29,997–999.

Megaridis, C. and R. Dobbins (1990). Morphologicaldescription of flame-generated materials. Combus-tion Science and Technology 71, 95–109.

Mei, R. and K. Hu (1999). On the collision rate ofsmall particles in turbulent flows. Journal of FluidMechanics 391, 67–89.

Meyer, C. J. and D. Deglon (2011). Particle colli-sion modeling – a review. Minerals Engineering 24,719–730.

Mokhtari, T. (1998). Studies fn the effects of shear oncolloidal aggregation and gelation using small an-gle light scattering. Ph. D. thesis, Sharif Universityof Technology.

Narsimhan, G. and E. Ruckenstein (2008). The brow-nian coagulation of aerosols over the entire range ofknudsen numbers: Connection between the stickingprobability and the interaction forces. J. Colloid In-terface Sci. 104, 344–369.

Pantina, J. and E. Furst (2006). Colloidal aggregatemicromechanics in the presence of divalent ions.Langmuir 22, 5282–5288.

Rothenbacher, S., A. Messerer, and G. Kasper (2008).Fragmentation and bond strength of airborne dieselsoot agglomerates. Particle and Fibre Toxicol-ogy 5:9.

Schlauch, E. C., V. Becker, M. Behr, and H. Briesen(2010). Finite element simulations of colloidal ag-gregates under shear flow conditions. V EuropeanConference on Computational Fluid Dynamics.

Selomulya, C. (2001). The effect of shear on floccula-tion and floe size/structure. Ph. D. thesis, Universityof New South Wales, Australia.

Sundaram, S. and L. R. Collins (1997). Collisionstatistics in an isotropic, particle-laden turbulentsuspension I. Direct numerical simulations. Jour-nal of Fluid Mechanics 335, 75–109.

Tsantilis, S. and S. E. Pratsinis (2004). Soft- andhard-agglomerate aerosols made at high tempera-tures. Langmuir 20, 5933–5939.

Tumolva, L., J. Y. Park, J. S. Kim, A. L. Miller, J. C.Chow, J. G. Watson, and K. Park (2010). Morpho-logical and elemental classification of freshly emit-ted soot particles and athmospheric ultrafine parti-cles using the tem/eds. Aerosol Sci. and Tech. 44,202–215.

Wengeler, R. and H. Nirschl (2006). Turbulent hydro-dynamic stress induced dispersion and fragmenta-tion of nanoscale agglomerates. J. Colloid InterfaceSci. 306, 262–273.

Xiong, C. and S. Friedlander (2001). Morphologicalproperties of atmospheric aerosol aggregates. Pro-ceedings of the National Academy of Sciences 98,11851–11856.

Zeidan, M., B. H. Xu, X. Jia, and R. A. Williams(2007). Simulation of aggregate deformation andbreakup in simple shear flows using a combinedcontinuum and discrete model. IChemE 85:A12,1645–1654.