Page 152 - Environmental Nanotechnology Applications and Impacts of Nanomaterials

138 Principles and Methods

the scattered intensity observed on a screen shows an ensemble of bright
and dark points, which are commonly referred to as speckles. If the
nanoparticles are immobile in the solution, the speckles are fixed on
the screen. However, as a result of Brownian motion, the intensity on the
screen fluctuates due to variations of the interference part of the scat-
tered intensity. These fluctuations are directly related to the diffusion
coefficient of the nanoparticles.
The signal is analyzed by a correlator to obtain the intensity auto-
correlation function Cstd 1 kIstdIst 1 tdl . When t 0, C(t) is simply
2
<I(t) >, but when t is sufficiently large, I(t) and I(t + ) are completely
2
uncorrelated and C(t) <I(t)> . The characteristic time required for the
2 2
autocorrelation function to go from <I(t) > to <I(t)> is related to the dif-
fusion coefficient of the nanoparticles. Therefore, for relatively dilute
suspensions of noninteracting and monodisperse nanoparticles the fol-
lowing relationship is true:
Cstd > e 2 t (26)

2
where Dq and D is the Stokes-Einstein diffusion coefficient
D 5 kT . For a polydisperse nanoparticle suspension, the autocorrela-
6p R
tion function is then:

2
Cstd 5 PsDde 2Dq t dD (27)
3
where P(D) is the normalized intensity weighted diffusion coefficient dis-
tribution function. Several algorithms exist to extract P(D) from the
correlation function. So, in principle, information about the particle size
distribution in a polydisperse suspension can be obtained.
For concentrated and/or interacting nanoparticles, the diffusion coef-
ficient is modified due to the hydrodynamic and thermodynamic (attrac-
tion and repulsion) interactions between the neighboring nanoparticles.
The motion of the nanoparticles is then strongly coupled and the cor-
relation function is no longer a simple exponential law. For typical par-
ticle size measurements, the suspension is diluted such that interparticle
interactions are negligible. Therefore, dynamic light scattering is not
suited for concentrated samples or complex mixtures where the parti-
cles are likely to be interacting with one another.

X-ray diffraction. The influence of particle size on the X-ray diffraction
pattern is quite important. The main size effect relates to the width of
the diffraction peaks for particles smaller than 50 nm (Figure 4.19).
The size of the crystallites can then be determined from the diffraction
pattern using the Sherrer formula (Eq. 1).