Supplementary Material
Dependence of initial cluster aggregation kinetics on
shear rate for particles of different sizes under turbulence
LYONEL EHRL, MIROSLAV SOOS, MASSIMO MORBIDELLI *
Institute for Chemical and Bioengineering, Department of Chemistry and Applied Biosciences,
ETH Zurich, 8093 Zurich, Switzerland
and MATTHÄUS ULRICH BÄBLER,
Institute of Process Engineering, Dept. of Mechanical and Process Engineering, ETH Zurich,
8093 Zurich, Switzerland
In the main paper we compared the results of the approach of an uniform aggregation efficiency1, 2
with literature data on doublet formation rates and cluster kinetics. Furthermore, we want to compare
the experimental data to the recent findings of Zaccone et al.3 who provided the rate of doublet
formation, K11 , analytically derived from the convective diffusion equation. For the case of zero
particle-particle interaction forces the following expression describes the transition from diffusion- to
shear-controlled aggregation:
K11 
8πDrp
2kBT
12π vr rp2

 12π vr rp2  
1  exp  

2kBT  


(1)
where vr is the relative velocity of the approaching particles. Using the definition of the Péclet number
in Eq. (1) one obtains the following expression:
*
corresponding author; e-mail: massimo.morbidelli@chem.ethz.ch; phone: +41-44-63-23033; fax: +41-44-63-21082;
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K11 
8πDrp
1 G rp
2Pe vr
(2)


v 
1  exp  2Pe r  
G rp  


To be able to compute K11 one needs to have an expression for vr . Using von Smoluchowskis’ rectilinear approach,4 neglecting the hydrodynamic interactions between the approaching particles, where
vr  G 2rp , one obtains:3
K11 
8πDrp
(3)
1
1  exp  4Pe  
4Pe 
However, as it was shown by van de Ven and Mason5 and Zeichner and Schowalter6 the effect of
hydrodynamic interactions is significant leading to a curvi-linear description of doublet formation. The
deviations from the recti-linear approach can be accounted for with an aggregation efficiency,
doublet  kPe0.18 , where the prefactor is a function of the London wavelength, L , and the particle
size.5 Thus we suggest to express the relative velocity of the particles according to
vr   doublet G 2rp  kPe0.18 G 2rp , where k  f  L  AH 12kBT 
0.18
.5 For the particle sizes
investigated here f  L  can be approximated to be equal to unity.5 Including this into Eq. (2) results
in:
K11 
abPe0.82
1  exp bPe0.82

(4)

where a  8kBT 3 and b  4  AH 12kBT 
0.18
. We are aware of the fact the term AH kBT in Eq. (4)
considers particle-particle interactions which were neglected in the derivation of Eq. (1), but we are
encouraged to take this approach, on the one hand, by the recent findings of Lattuada and Morbidelli7
on the use a functional form similar to Eq. (3) to describe K11 also in the presence of particle-particle
interactions, and on the other hand, by the fact that in experimental systems AH kBT is usually
constant. Consequently, from Eq. (4) n as a function of Pe was determined by calculating
1   log K11  log Pe  1   K11 Pe  Pe K11  , resulting in the expression
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n  0.18 
0.82bPe0.82
1  exp bPe0.82

(5)

which is plotted as a solid line in Figure 5 of the main paper, for Hamaker constants of (solid)
polystyrene, (dashed) silica, and (dash-dotted) titania. For large values of Pe the curvi-linear solution
is recovered, which corresponds to the experimental data of the literature and this work, where the rate
of aggregation is non-linearly dependent on G with n  0.18 . However, for small values of Pe a
transition from purely shear-induced to diffusion-affected aggregation is observed. A change in the
Hamaker constant due to different material properties shows little effect on the position of the curve.
This indicates little sensitivity of the interplay between Brownian and shear-induced aggregation to
attractive inter-particle forces, which is in contrast to the tremendous effect of repulsive inter-particle
forces on doublet formation rates under shear as shown by Lattuada and Morbidelli7. The much steeper
decay of n with an increase in Pe could be either an effect of the assumptions taken in the derivation
of Eq. (1)3 or due to the fact that for shear-induced aggregation doublet formation rates are not
capturing the dynamic characteristics of the whole process, as it is the case for static aggregation
kinetics.8-10 However, one can conclude that by combining the simple mathematical expression
provided by Zaccone et al.3 under the assumption of zero particle-particle interactions with the proper
value of the relative velocity of particles the cross-over from shear- to diffusion-controlled aggregation
can be captured.
For this reasons, we conclude that proper aggregation efficiency models for cluster-cluster
aggregation should exhibit a dependency of the exponent n on Pe as observed by measurements and
listed in Table 2.
1.
2.
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3.
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von Smoluchowski, M. Versuch einer mathematischen Theorie der Koagulationskinetik
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