Environmental Engineering Reference
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0 (h) = [6(h/a)
2
+ 13(h/a) + 2]/[6(h/a)
2
+ 13(h/a)]
(Eq. 15.84)
which corrects for the hydrodynamic resistance (interaction) between the two
approaching particles. Tadros (2007) proposed the following equation for simplifying
logW = constant - 2.06 X 10
9
(Ry
2
/Z
2
)logC
(Eq. 15.85)
+ l]
(Eq. 15.86)
where C = electrolyte concentration, Z = valency, and TO = surface potential.
For dilute dispersions, the aggregation kinetics of NPs can be determined by
time-resolved dynamic light scattering (DLS) experiments (Heidmann et al., 2005; Chen
and Elimelech, 2006). Using DLS the average hydrodynamic radius rj, of NPs can be
measured as a function oft, starting with a well-dispersed system at t = 0:
r
h
= k
B
T/6
n u
D
(Eq. 15.87)
where D = the intensity-weighted average translational diffusion coefficient, which can
be extracted using the methods of cumulants from the intensity autocorrelation functions
(Tadros, 2007). The early-stage aggregation kinetics is expressed as the initial rate of
increase in rj, with time, t, as measured by DLS as follows:
(dr
h
(t)/dt)
t
^
0
« k
n
,
slow
N
0
(Eq. 15.88)
where NO = the initial (primary) NP concentration. kn
jS
i
o
w
= the initial aggregation rate
constant = the rate coefficient of slow flocculation (in the presence of an energy barrier,
Oimax). The initial rate of increase in rh(t) can be obtained through a consistent approach
by determining the initial slope of the linear regime of the aggregation curves (e.g., up to
the point where the rj, reaches 1 .25 rh(
t
= o))- The aggregation attachment efficiencies a (=
1/W, Tadros, 2007) at different electrolyte concentrations can be calculated as
0
/
N
o]/[(dr
h
(t)/dt)^
ofast
/N
Wast
]
(Eq. 15.89)
where the terms with subscript "fast" refer to favorable (nonrepulsive) aggregation
conditions. Eqs. 15.87-89 have been widely used in studies on aggregation kinetics of
NMs (Heidmann et al., 2005; Chen and Elimelech, 2006).
For concentrated suspensions of NMs, there may be two (or more) phases of NP
aggregation with the NPs rapidly aggregating to form discrete, micrometer-sized
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