Page 298 - Environmental Nanotechnology Applications and Impacts of Nanomaterials

Nanoparticle Transport, Aggregation, and Deposition 283

can calculate d by plotting Eq. 32 and solving for the inverse slope of
w
a log/log plot of t (x) versus x [e.g., logsxd ~ d 1 logstd ].
c
w
Due to this steric contribution, as particle size approaches that of the
pore size particles diffuse within a disordered structure in a constrained
fashion. In bacterial biofilms, Lacroix-Gueu et al. [109] showed the
enabled anomalous diffusion of latexes and bacteriophages with 55 nm
radius. In 1.5 to 2 weight percent agarose gels, the steric obstructions
appear with solute sizes above 10 nm, and a critical size is measured
around 70 nm [100, 110]. However, the critical size is logically inversely
correlated to the gel density. Labille et al. [108] showed with the same
agarose gel that in a gel/solution interface, the fiber density is locally
increased from 1.5 to 5 percent by weight in a 100- m thickness inter-
phase layer, through which the maximum size of diffusing particles is
decreased to around 50 nm radius. The diffusion motion of macromole-
cules has also been measured in intracellular cytoplasm, where dextran
with molecular weight up to 2 106 Da (~ 45 nm radius) undergoes
impaired diffusion [111].

Electrostatic interactions. Most gels, flocs, and biofilms encountered in
aquatic and soil environments are mainly composed of polysaccharides
and humic substances. Both of these highly reactive components are neg-
atively charged in a large pH range due to the presence of acidic groups
in their chemical structure [89, 112]. This induces a negative surface
charge to the global diffusing media, which is characterized as follows
by the Donnan potential , which is the average difference in potential
between gel and bulk water:

kT 21 r
c 5 sinh (33)
zF 2zFc
where is the charge density of the gel; c and z are respectively the molar
concentration and charge of the electrolyte in the external bulk solution;
and F is the Faraday constant. The diffusion motion in the gel of charged
nanoparticles is thus affected by an electrostatic contribution that can
be described by a Boltzmann distribution:
[A] Z Fc
A
P
p 5 5 expa2 b (34)
[A] w kT
where [A] is the concentration of particles A in gel pores exclusively con-
P
trolled by electrostatic interactions and Z is electrical charge of the par-
A
ticles. Figure 7.32 exemplifies the effect of a negatively charged agarose
gel on the diffusion motion of a positively charged solute. The anionic gel
charge is neutralized by protonation at pH < 3.5, and is screened at
pH > 9 by increased ionic strength required for pH adjustment. The