Page 260 - Environmental Nanotechnology Applications and Impacts of Nanomaterials

Nanoparticle Transport, Aggregation, and Deposition 245

determined by taking the ratio of rates observed at ionic strengths below
and above the CCC. Rates may be determined for either monodisperse or
polydisperse suspensions, though the latter case is exceedingly more com-
plex as a range of particle sizes, shapes, and interaction potentials must
be considered. As an alternative to extracting initial aggregation rates
from measurements of mean aggregate diameter over time, measure-
ments of particle size distribution over time can be compared with those
calculated from a particle population model. In this case, W is treated as
a fitting coefficient. Such an approach is well suited to polydisperse sus-
pensions and has been used to track nanoparticle aggregation [16].
The classical kinetic approach of Von Smoluchowski can be adapted to
a particle population model that may be used to assess the aggregation
behavior of nanoparticles. Apopulation balance describing the kinetics of
aggregation and the evolution of the particle size distribution over time
can be written as a system of differential equations [39]. The assumption
that aggregation occurs in sequential steps of transport and attachment
is embodied in these equations by multiplying rate constants that describe
particle transport by stickiness coefficients that describe attachment. In
the case of an irreversible aggregation, the evolution of the number con-
centration, n , of a particle in size class k can be written as follows:
k
dn k 1
5 a bsi, jdn n 2 an k bsi,kdn i (8)
i j
dt 2 i1jSk i
where is the attachment efficiency (or stickiness coefficient, equal to 1/W),
and (i, j) are the collision frequencies (transport) between particles or
aggregates of classes i and j. The population balance can be modified to
account for factors such as particle sedimentation and particle breakup.
Approaches to computing the collision frequency kernel (i, j) typically con-
), shear-
sider three principal collision mechanisms: Brownian diffusion ( br
induced collisions ( ), and differential settling ( ).
sh
ds
Two ideal cases have been presented in the literature for calculating
particle collisions by each of these mechanisms. In one case, the recti-
linear model, particles are assumed to have no effect on the fluid they
are suspended in, as though all fluid passes through the aggregate or
particle as it diffuses, settles, or is carried by the fluid. In contrast, the
curvilinear model [40] considers the effect of creeping flow around par-
ticles and compressed streamlines as two particles approach one
another. The curvilinear model predicts many fewer collisions between
particles than does the rectilinear model since fluid must be displaced
before particle contact occurs. The curvilinear model with particle
transport defined primarily by Brownian diffusion would appear to be
appropriate for describing fullerene transport early during the aggre-
gation process, particularly if the resulting aggregates are compact
and highly ordered. However, if aggregation leads to the formation of