Page 148 - Environmental Nanotechnology Applications and Impacts of Nanomaterials

134 Principles and Methods

where S(q) is the structure factor accounting for the correlation between
the nanoparticles.

1 siqsr j 2r i dd
Ssqd 5 1 1 Q e R (19)
N a i a j2i
A detailed quantitative analysis of the structure factor goes beyond
the scope of this book. However, a qualitative description of the struc-
ture factor can give useful information on the state of the nanoparticle
dispersion. The value of S(0) is proportional to the average mass of the
nanoparticle aggregates and the colloidal liquid compressibility. For a
dilute suspension, S(q) 1. For stable nanoparticle suspensions, the
limit of S(q) when q approaches zero is less than 1. For unstable
nanoparticle suspensions, S(q) is larger than 1 when q approaches zero.

Limiting behaviors of the form factor. In the limit of forward scattering and
for dilute nanoparticles suspensions, the form factor can be approxi-
mated as an exponential:

2sqR 0 d 2
a b (20)
Isqd< fV part e 3
where R is the radius of gyration of the nanoparticle. This regime of
G
forward scattering is called the Guinier regime and is valid for qR < 1.
G
In the limit of very large q, and provided that a perfect interface exists
between the nanoparticles and the solution (i.e., no surface rugosity),
then the following limit is observed:

2
2ps rd S
lim q1` Isqd 5 4 (21)
q V
where is the scattering length density contrast between the nanopar-
ticles and the solvent, S, is the total surface area of the nanoparticles,
and V is the scattering volume. This approximation allows the specific
surface area of a nanoparticle to be measured. When the suspension is
concentrated or if the solution is bicontinuous, the same formula holds.
This large angle regime is called the Porod regime. The specific surface
4
of the particles is conveniently extracted using a I(q)q vs. q plot as
shown in Figure 4.3.
Example SAXS experiment. As an illustration of these principles, let us con-
sider an example of the information that may be obtained through small
angle X-ray scattering (SAXS) by a suspension of CeO nanoparticles.
2
Our goal is to characterize concentration, size, and shape of the nanopar-
ticles in the suspension. The initial nanoparticle suspension is made by
(NO ) (H 0)
peptisation of 38 g of a powder with raw formula CeO 2 3 0.5 2 x
mixed with 30 g of poly acrylic acid (C H O ) and a molar mass of
3
2 n
4