Biomedical Engineering Reference
In-Depth Information
where A
e
is the amplitude scattered by one electron,
ρ
(
r
) is electronic density and the
product
) dV is the number of electrons inside the scattering volume dV.
In light scattering experiments, (
2.5
) becomes
ρ
(
r
e
iq
r
dV
dA
ðqÞ¼
C
ρðrÞ
;
ð
2
:
6
Þ
where C is a coef
cient which depends on detection conditions and
ρ
(
r
) is the polar-
izability of an elemental volume dV.
Equations (
2.5
) and (
2.6
) show that the amplitude of the scattered wave is the Fourier
transform of the functions
ρ
r
q
(
). The total amplitude A(
) scattered by volume V is
ρ
r
obtained by integration of these equations. If the function
(
) had a constant value in
the whole volume, the integral would be a Dirac function
δ
(
q
): no wave would be
scattered apart from the incident direction
q
= 0. Consequently, any light scattered in
direction
θ
with scattering vector
q
is due to local
fluctuations of
ρ
(
r
).
The mean intensity I(
q
) of scatter at scattering vector
q
is the average of the product,
refer to the average over all orientations and all
positions of the particles inside the scattering volume:
A(
q
) A*(
q
)
. Here the brackets
〈
〉
〈〉
I
ðqÞ¼h
A
ðqÞ
A
ðqÞi:
ð
2
:
7
Þ
Particles (atoms, molecules, colloids) interact through intermolecular forces which, as a
first approximation, depend on the relative distances between the centres of the particles
(for example, steric repulsion, double layer repulsions in colloids and long-range van der
Waals attractions). Assuming that the interaction between a pair of particles is isotropic in
space (i.e. it depends only on the distance between particles and not on their relative
orientations), a full statistical physics treatment allows the derivation of an expression for
the intensity I(
q
) of the radiation scattered by N particles in direction
u
s
. The function I(
q
)
can be separated into two factors:
I
ðqÞ¼
I
ð
q
Þ¼
NI
1
ð
q
Þ
S
ð
q
Þ;
ð
2
:
8
Þ
where I
1
(q) is the intensity scattered by a single particle (such as a colloidal particle,
polymer coil or micelle) or the
'
'
form factor
measured in very dilute solution, and S(q)
is the so-called
ects the relative positions of the particles
related to their interaction potentials. The structure factor is equal to 1 when the
solution is extremely dilute and the scattering by N particles is just the addition of
the intensities scattered by the individual particles. In more concentrated solutions, the
structure factor becomes increasingly important and re
'
structure factor
'
,whichre
ects the so-called
'
short-range
order
eld
repulsion core of spherical particles over large distances, beyond two to three times the
centre-to-centre distance, but the short-range order is important and re
'
in the solution. Brownian motion does not allow ordering of the hard
ects the ther-
modynamic stability of the solution. The second virial coef
cient can be derived from
this measurement.
When scattering techniques are used to investigate the local structure of physical gels,
these two factors are both very important.
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