Fate and Transport of Nanomaterials in Aquatic Environments - Nanotechnologies for Water Environment Applications - page 530

aggregates (1 s regime) followed by subsequent chain-like gelation (2 n regime). In the

1 s t regime, the aforementioned DLS methods can be used; in the 2 n d regime, the

hydrodynamic radius of the aggregates cannot be measured directly with DLS due to
multiple scattering (Phenrat et al., 2007). Because of the existence of the two regimes,
Phenrat et al. (2007) reported that, for the concentrated dispersions (190-1130 mg/L),
the sedimentation curves of RNIP have three distinct regions: (a) aggregation I in which
the NPs aggregate to form the chain-like aggregates but largely remain suspended; (b)
sedimentation I in which the chain-like aggregates gel to reach a critical size at a critical
time t^t so that sedimentation of the chain-like aggregates becomes rapid; and (c)
sedimentation II in which the sedimentation rate is low because the sedimentation is
mainly contributed by the sedimentation of NPs that does not form critical size chain-
like clusters. At an initial concentration < ~7.5 mg/L, the aggregates never reach a
critical size and are only slowly sedimented from the solution. Similar results are also
true for magnetite and hematite dispersions.

15.5.1.4 Fractal Flocculation

The short-range forces governing the interactions between spherical NPs
influence not only the kinetics of aggregate formation, but also the structure and
resulting fractal dimension of the NP aggregates (Tadros, 2007). For example, Chen et
al. (2006) reported the alginate-coated hematite of an average diameter of 75 nm
aggregated in the presence of 391.7 mM NaCl at pH 5.2 (the favorable aggregation
conditions corresponding to fast aggregation) with linear dimensions of about 1 |im.
Lecoanet et al. (2004) reported that the anatase particles of 40 nm aggregated with
formed clusters having an irregular shape with a mean size of 198 nm, while n-Ceo
aggregated with formed clusters having highly regular shape with faceted edges and a
mean diameter of 169 nm. When rapid flocculation of small particles occurs, dendrites
(i.e., snowflake-like structures, known as fractals) form, resulting in particles that are
much larger than those predicted from the number of small particles using Euclidean
geometry and assuming a closely packed material.

The fractal dimension D is used to describe the fractal volume by accounting for
differences in particle shape (Logan, 1999):

Volume ocng (Eq. 15.90)

where n p = the concentration of particles in the aggregate. The magnitude of D is in part
a function of the way the aggregate is grown. When particles are added one at a time
(monomer-cluster), D ranges from 3 for reaction-limited conditions to 2.5 for diffusion-
limited conditions. In rapid aggregation, when cluster-cluster interactions become
important, D ranges from 1.7-1.8 for diffusion-limited conditions to 2.09 for reaction-
limited conditions (Logan, 1999), and to 2.1-2.2 when adhesion to the cluster is the rate-