Page 283 - Environmental Nanotechnology Applications and Impacts of Nanomaterials

268 Principles and Methods

where is the velocity gradient at the collector surface and v is the kine-
matic viscosity of the fluid. The lift force arises from the different pres-
sure forces acting on the top and bottom of the particle due to the velocity
gradient. For nanoparticles, F is negligible in comparison to the adhe-
L
sive force and may likely be omitted [62]. The result is a torque exerted
on the particles due to the hydrodynamic drag given by:

T 5 1.399a F (26)
D
p D
where the leading coefficient (1.399) indicates that the drag force acts
on the deposited particle at an effective distance of 1.399a from the sur-
p
face [62]. In other words, the hydrodynamic torque acts over a lever arm
of l 0.339a .
p
y
The adhesive forces are a result of physicochemical interactions occur-
ring between the two interacting bodies. The favorability or adhesion
strength may be quantified in terms of the free energy of adhesion (W )
A
using an XDLVO type approach for determining the free energy at con-
tact. The adhesion force (F ) may then be determined using different
A
scaling models, such as the Johnson-Kendell-Roberts (JKR) [66] and
Derjaguin-Mullen-Toporov (DMT) [67] models. This adhesive force
resists the torque exerted by the drag force, and is manifested as a
torque acting over a specified lever arm of l :
x
5 F l (27)
T A A x
The value of l is taken to be equal to the radius of the contact area
x
between the collector and the retained nanoparticle. The size of the con-
tact area is determined by a number of different parameters, the most
significant of which are the elasticity of the interacting bodies, particle
size, and surface roughness.
Elasticity describes the stiffness or malleability of a material. As
elasticity increases, the contact area will also increase in size as the sur-
faces can deform and wrap around each other [68]. As is illustrated in
Figure 7.23, the contact area also decreases both on approach and at con-
tact with decreasing particle size. Nanoparticles interact with a smaller
region of the collector surface as they approach and ultimately make con-
tact with the collector surface. Surface roughness may result in either
an increase or a decrease in the contact area depending on particle size
and the density of asperities on the collector surface. Roughness is likely
to result in an increased contact area between the collector and the
nanoparticle, resulting in an increase in the adhesive force.
Typically, fluid drag or hydrodynamic shearing forces are significant
only for particles larger than several hundred nanometers when depo-
sition occurs in the primary minimum [69]. This is due to several factors,