Environmental Engineering Reference
In-Depth Information
The macroscopic approach overcomes the summation assumption by considering
the macroscopic electromagnetic properties of the medium (Lifshitz, 1956) where
atomic structure is neglected and large bodies are treated as continuous media and
forces are derived in terms of the bulk properties such as dielectric constants and
refractive indices. However, the use of the macroscopic approach is limited by the
computation required and the lack of appropriate dielectric data. Additional details
about these two approaches and calculations of Hamaker constant can be found
elsewhere (Bergstrom, 1997; Elimelech
et al.
, 1995a ; Israelachvili, 1972 ).
4.5.1.2
Double Layer Interaction
Colloidal particles often carry an electrical charge and therefore attract or repel
each other. When two like-charged particles approach each other, their electrical
double layer starts to overlap, resulting in a repulsive force which opposes further
approach. For identical particles, sphere-sphere double layer interaction energy can
be given by Equation 4.6. There are many expressions available based on various
assumptions for sphere-sphere double layer interaction energy and readers are
referred to the literature for more details (Bell
et al.
, 1970 ; Carnie
et al.
, 1994 ;
Genxiang
et al.
, 2001 ; McCormack
et al.
, 1995 ; Sader
et al.
, 1995 ; Stankovich and
Carnie, 1996 ).
( )
2
R
kT
ze
Vh
(4.6)
( )
=
32
πε
γκ
2
exp
(
−
h
)
R
For small values of surface or zeta potential (
ζ
), this simplifi es to:
( ) =
2
πζ
(
)
Vh
2
eR
exp
−
κ
h
(4.7)
R
where
is dimensionless
functions of the surface potentials, k is the Boltzman constant, T is the absolute
temperature (Kelvin), h is the surface-surface separation between particles (m), e
is the electron charge and
ε
is the permittivity of the medium, R is the particle radius,
γ
is the inverse of Debye-Huckel screening length (m
− 1
).
Equation 4.7 is applicable only if
κ
R. For the general case of elec-
trolyte solutions containing a number of dissolved salts,
κ
R
>
5 and h
<<
κ
is defi ned by:
∑
2
2
enz
kT
ii
(4.8)
κ
=
ε
where n is the number concentration of ion i in the solution.
Inserting numerical values appropriate to aqueous solutions at 25 °C and con-
verting the ion concentration into molar terms gives:
κ =×
(
)
∑
c
ii
(4.9)
232
.
10
9
2
where c the concentration of ions expressed in mol l
− 1
and z the valency of the ions.
The length 1/
is known as the thickness of the diffuse layer. Equation 4.9 shows
that the increase in ionic strength results in a decrease in the thickness of the diffuse
layer and a consequent decrease in the repulsive interactions among particles.
Typical values of the diffuse layer thickness, 1/
κ
κ
, are in the range 1- 100 nm.
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