Page 251 - Environmental Nanotechnology Applications and Impacts of Nanomaterials

236 Principles and Methods

Electrostatic interactions. Hogg et al. [11] derived an expression for the
electrostatic interaction energy between two surfaces under the
assumption of constant surface potential. When modeling electrostatic
interactions between surfaces, a theoretical constraint of constant sur-
face potential or constant surface charge is generally made [7]. In
reality, the actual condition lies between these two experimental
extremes. The electrostatic interaction energy per unit area between
a spherical particle and a flat surface decays with separation distance
according to:

2
2
El
U 123 shd 5 pe e a c2z z lna 1 1 e 2kh b 1 sz 1 z dlns1 2 e 22kh dd (3)
r 0 p
1 3
1
3
1 2 e 2kh
is the dielectric permittivity of the suspending fluid; is the
where 0 r
Debye constant; and and are the surface potentials of the interacting
1
2
surfaces. The inverse of the Debye constant is a measure of the diffuse
ionic double layer thickness that surrounds charged surfaces in aque-
ous systems. For this reason it is often referred to as the inverse Debye
screening length and is determined according to the following equation
for z-z electrolytes:
2
e gn z 2
i i
k 5 (4)
Å e e kT
r 0
is the number concentration of ion i
where e is the electron charge; n i
in the bulk solution; and z is the valence of ion i.
i
According to Eq. 3, the electrostatic interaction energies decay expo-
nentially with distance and are a function of both the separation dis-
tance (h) and the Debye length (1/ ) [7, 12]. Moreover, electrostatic
repulsion is predicted to decrease with decreasing particle size (Figure 7.3),
and may therefore decrease barriers to nanoparticle aggregation. The
distance from charged surfaces over which repulsive interactions occur
is often similar to the size nanoparticles as reflected in the Debye length.
The variability of 1/ is dependent on the ionic strength of the solution and
the valency of the ionic species present [13]. For instance, in a 100 mM
NaCl solution 1/ is equal to roughly 1.0 nm, while for a 0.01 mM NaCl
solution 1/ is approximately 100 nm. Thus, the distance over which elec-
trostatic interactions exert their influence changes based on the ionic
strength and types of ions in the solution. Therefore, it becomes impor-
tant when considering EL interactions in describing nanoparticle behav-
ior in water to consider the salt concentration and valence of the ions
present. Furthermore, the thickness of the diffuse double layer has
specific consequences unique to nanoparticles as they may be present