Page 261 - Environmental Nanotechnology Applications and Impacts of Nanomaterials

246 Principles and Methods

porous aggregates, neither the rectilinear nor the curvilinear model
yields satisfactory results compared with laboratory observations. An
intermediate model [41] is required that considers the entire range of
cases where the effect of fluid flow around each aggregate as well as
the potential for fluid to pass through the aggregate are taken into
account. Closed-form approximations have been proposed [42] for cal-
culating the collision frequency kernel as a function of the ratio of drag
force on a permeable aggregate to that on an impermeable sphere, ,
and the fraction of fluid that passes through an isolated aggregate, :

2kT 1 1
s d 5 a 1 bsr 1 r d (9)
b Br i,j i j
3m i r i j r j
3
1
s d 5 a2 r 1 2 r b G (10)
b sh i,j i i i i
6
2
*
*
b s d 5 pa 2 r 1 2 r b `u 2 u ` (11)
ds i,j
i i
i
j
i i
where r is the radius of the aggregate or particle, is the dispersing
medium viscosity (0.000890 kg/m sec for water), G is the mean shear rate
1
(s ), and u* is the settling velocity of the particle or aggregate.
Assuming that the transport step of aggregation can be adequately
expressed by calculated values for the collision kernel, ij , then the sta-
bility ratio can be determined by adjusting its reciprocal, , so that cal-
culations of particle size distributions over time correspond with the
observed particle size distributions.
For a suspension of nanoparticles all having the same size, assuming
that collisions between particles j leads to irreversible attachment with
an efficiency of , the decrease in number concentration of unit parti-
cles of size j in suspension can be estimated as:
dn j n j 2
52ab a b (12)
jj
dt 2
where is the specific particle collision frequency, and n is the number
jj
j
concentration of particles j. If all particle contacts result in aggregation,
n
dn j j 2
then (1/W) 1 and 52b A B 5 K . Compared to micron-sized
dt ii 2 jj,fast
particles, nanoparticles will have considerably smaller inertia and
approach velocities prior to impact with another particle [16]. This results
in extended times in the interaction fields of the other particle, meaning
that nanoparticles are more likely to be deflected away as a result of
repulsive forces. This would also suggest that nanoparticles do not
“collide” in the literal sense, but rather slow down to a velocity of zero