Environmental Engineering Reference
In-Depth Information
4.5.2
Stability Criteria
According to DLVO theory, various parameters can affect colloidal stability, such
as ion type and concentration, the value of
potential and the particle size.
According to Equation 4.9, an increase in the ionic strength results in a decrease
in the diffuse layer thickness and, therefore, a decrease in the double layer repulsive
force. Polyvalent electrolytes induce larger decreases in the diffuse layer thickness
than monovalent electrolytes and consequently induce greater aggregation.
According to Equation 4.7, the electrostatic repulsive force is proportional to the
square of
ζ
potential, that is a doubling of the zeta potential results in quadrupling
the repulsive force, and so it is a key parameter in determining the stability of col-
loids. However, for some environmental colloids such as humic sbstances, the
physical meaning of
ζ
has been questioned (Duval
et al.
, 2005 ). According to
Equation 4.4 and Equation 4.6, both attractive and repulsive forces are propor-
tional to the particle size. At small sizes, the value of V
T
(total interaction energy)
is directly proportional to the size. However, at large sizes the value of V
T
has a
more complicated variation. In all cases, electrostatic stability increases with
increasing particle size.
ζ
4.5.3
Aggregation Kinetics
For dilute colloidal system where only binary collisions are assumed to take place,
the kinetics of particle coagulation due to Brownian motion can be described by
the Smoluchowski rate equation (Holthoff
et al.
, 1996 ):
dN
dt
1
2
∞
∑
∑
z
(4.10)
=
kNN
−
N
k N
ij
i
j
z
iz
i
ijz
+=
i
=
1
where t is the time, N
z
is the concentration of z-fold aggregates and k
ij
is the rate
at which i-mers particle bind to j-mers, following the diffusion of particles toward
each other. According to Smoluchowski, the aggregation rate constant for the for-
mation of dimers from monodisperse suspension of monomers can be given by:
kk
kT
s
8
B
(4.11)
==
η
2
11
3
where k
B
is Boltzmann constant, T is the absolute temperature and
is the viscosity
of the fl uid. This constant k
11
is thus independent of the size of particles. Considering
the van der Waals forces and the hydrodynamic interactions, the coagulation rate
can be can be expressed as:
η
−
1
∞
(
)
(
)
h
hR
β
Vh
kT
∫
A
kk
=
2
exp
dh
(4.12)
11
s
2
(
)
+
2
B
0
where
(u) is the correction factor for the diffusion coeffi cient (Overbeek, 1982),
h is the distance between particle surfaces and R is the particle radius. Equation
4.12 is characteristic of the fast or diffusion limited aggregation regime. In the
regime of slow or reaction limited aggregation, additional repulsive forces due to
electrostatic interactions may prevent the particles from aggregating. In such a
β
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