Fate and Transport of Nanomaterials in Aquatic Environments - Nanotechnologies for Water Environment Applications - page 484

Diffusion coefficients can be determined experimentally using different methods
(e.g., free boundary methods, the porous plug method, dynamic light scattering) (Dunlop
et al., 1977) or calculated using a variety of correlations, including Stockes-Einstein
equation, Poison equation, Hayduk-Laudie correlation, and Nernst-Haskell equation, etc.
(MWH, 2005). Currently, sufficient information is not available on diffusivity of
different NMs or NPs in the aquatic environment under different conditions (e.g., pH,
ionic strength, presence of natural organic matters). Stockes-Einstein equation is
recommended as it works for colloidal and large, round molecules in any dilute solution

D = k B T/(3 n v. d p )

(Eq. 15.23)

At a fixed temperature (e.g., 25°C), particle diffusivity is a simple function of size, that

is, D (cm 2 /s) = 4.9 x 10 " 6 /d p (nm). Given the range of 1 to 1000 nm in NPs' diameter,

the variation in diffusivities of NPs is from 10" 6 to 10" 9 cm 2 /s, which is different from
round molecules (Table 15.2). The average diffusivity of NPs increases with decreasing
size, and therefore, show delayed sedimentation in the earth's gravitational field, which
translates into potentially increased lifetimes for NP impurities at low concentration. In
the presence of larger NPs, as with the wide size distribution in aerosols such as smoke,
the highly diffusive character of NPs may lead to faster agglomeration or impaction on
the larger particles.

(crnVs) 1 1

Table 15.2 Diffusivity as a function of NPs' diameter or molecular weight.

(nm) b

(Dalton) 1 1

Diffusivity Substance MW Diameter Diffusivity

(cm 2 /s) a

10 1
10 2
10 3
10 4
10 5
10 6

2.2x1 O' 5
7.0x1 0' 6
2.5xlO' 6
l.lxlO' 6
5.0x1 0' 7
2.5xlO' 7

4.9x10-"
4.9xlO' 7
9.8xlO' 8
4.9xlO' 8
9.8xlO' 9
4.9xlO' 9

0.29
0.62
1.32
2.85
6.2
13.2

a D (cm 2 /s) = 4.9 x 10 " 6 /d p (nm). b MW = molecular weight (Perry and Chilton, 1973).

Flocculation of NMs in Water (Classical Models). NPs and NMs tend to
aggregate to form clusters, which may behave as conventional suspended particles, and
can be described with classical flocculation models. Smoluchowski (1917) developed
rectilinear collision models for spheres with many assumptions (Thomas et al., 1999).
The rate of NP attachments ry can be described as follows:

i-jj = ap^inj (Eq. 15.24)
where ry = rate of attachment between i and j NPs, collisions/L 3> T (#/cm 3> s); a =
collision efficiency factor (attachments per collision, 0 < a <1); (3y = overall collision