Environmental Engineering Reference
In-Depth Information
where OT = the total interaction energy between two particles; OEDL = the electrostatic
energy due to the electrical double layer between two particles; and Oy = the London-
van der Waals energy of attraction between two particles. The traditional DLVO theory
utilizes the Derjaguin approximation, requiring that the characteristic thickness of the
EDL (Debye length) is smaller than the radius of curvature of the particle. In addition,
the distance between the particle and surface must be much less than the size of the
particle. For NPs, these approximations are not valid due to their small size. Usually,
OEDL and Oy depend on NPs' geometries and their arrangement. Here, only calculations
of OEDL and Oy in two cases will be introduced.
First, for calculating interaction energies between two NPs, the following
equations derived for small charged molecules or ions (Leckband and Israelachvili,
2001) should be used:
(Eq. 15.73)
where zi and Z2 = the valence of NP 1 and NP 2, respectively; e is the electron charge
(1.60219 x 10"
1
9
C); K = the inverse Debye screening length; D = the distance of closest
approach between the two surfaces; so = the permittivity of vacuum; s
r
= the relative
dielectric constant of the medium; R = the center to center distance between the NPs; a =
the radius of the sphere; and
4V=-^F
(Eq. 15.74)
where CVDW = the van der Waals constant which depends on the optical properties and
geometry of the interacting bodies (see eq. 15.78).
Second, for calculating the interaction energy of a sphere (e.g., a NP) and a flat
plate (e.g., porous media), Guzman et al. (2006) suggested using the following equation
to calculate OEDL and Oy:
<S>
EDL
= Jre
r
£
0
(¥
s
2
+ ¥
p
2
) J
0
a
{-coth[
K
(D + a - aVl - (r/a)
2
)] + coth[
K
(D + a + aVl - (r/a)
2
)] + [2¥
s
¥
p
/( ¥
s
2
+
¥
p
2
)][csch[K(D + a - a,/! - (r/a)
2
)] - csch [K(D + a +
ajl -
(r/a)
2
)]]}rdr
(Eq. 15.75)
where SQ = the permittivity of vacuum;
e
r
=
the relative dielectric constant of the
medium; K = the inverse Debye screening length (Debye-Hilckel parameter); Ys and Yp
= the surface potentials of the sphere and the plate, respectively;
a =
the radius of the
sphere;
D =
the distance of closest approach between the two surfaces; and r = the
distance (0 < r < a). Eq. 15.75 can be used to calculate interaction energies between NPs
and the surface of porous media, which is assumed to act as a flat plate.
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