Environmental Engineering Reference
In-Depth Information
The surface potential of the NP or the flat plate in water as a function of pH can
be calculated with the Nernst equation:
where
Yj =
the surface potentials of material i (e.g., i = s for NP or = p for the flat plate
in eq. 15.75); ks = Boltzmann's constant; T = temperature; and pHzpci = the pH at the
point of zero charge. The van der Waals interaction energy can be obtained from:
n
'
v
\D+2a/
J
O
=
_^H
[
±+^_ + i
n
('^_)]
6
L
D
D+2a
(Eq. 15.77)
v
where AH = the Hamaker constant, which can be defined for a van der Waals body-body
A
H
=
K
2
C
V
v
wPl
p
2
(Eq. 15.78)
where pi and p2 = the number of atoms per unit volume in two interacting bodies, that is,
Pi
= pL
A
/m
a
(Eq. 15.79)
where p = the particle density of body i; LA = the Avogadro number; and m
a
= the molar
mass of body i. The Hamaker constant ignores the influence of an intervening medium
between the two particles of interaction, which is given by the Lifshitz theory for two
bodies 1 interacting across a medium 2 as follows (Leckband and Israelachvili, 2001):
A = © kBT(^ff)
2
+
^<g^£
(Eq. 15.80)
where si, 82 and ni, n2 = the static dielectric constants and refractive indices of the two
phases (e.g., 1 for NP, and 2 for water); and I = their ionization potential, which is close
to 2 x 10" J for most materials. In the presence of a medium (solvent), the Hamaker
constant A^i of material 1 (e.g., NP) dispersed in a medium (e.g., water) of Hamaker
constant A22 is give as follows (Israelachvili, 1991):
Am * (4i -
A
L)
2
(Eq. 15.81)
where AH and A22 = the Hamaker constants of NPs and water interacting in a vacuum,
respectively. For non-metallic solids and liquids interacting in vacuum or air (82 = n2 =
1), their Hamaker constants are typically in the range of (5-10) x 10"
2
0
J, while for
interactions in a liquid medium such as water, the Hamaker constants are typically an
order of magnitude smaller and in the range of (0.5-1 .5) x 10" J. For example,
AH ~
0.8
x 10"
2
0
J for lipid bilayers, (1.0-1.5) x 10"
2
0
J for proteins interacting in water or salt
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