Chemistry Reference
In-Depth Information
This correlation function has the form
16
q
6t
t
L
l
þ
3
q
6t
t
q
cosh
l
6t
t
q
6t
t
g
ð
1
Þ
ð
t
Þ
h
i
þ
3
h
i
ð
1
Þ
sinh
l
1
þ
8t
3
t
for t
{
t
, where
t
¼
(Dk
2
)
1
, and D is the particle diffusion coefficient,
k
¼
2
p
n/
l
the wave vector of the light, n the refractive index of the medium, and
L the thickness of the sample being measured. This form of correlation function
is only valid when L c l* and t
{
t
, which is true for all of our experiments.
The parameter l* is the photon transport mean free path, defined as the length
scale over which the direction of the scattered light has been completely
randomized. Practically, it is essentially a turbidity parameter and is directly
related to the total scattered light intensity of the system. The parameter l*is
not measured as part of the measurement of the correlation function, but it can
be determined from the long-time average of the intensity of the scattered
light.
17
If l* is known, it can be seen that the only unknown quantity in
Equation (1) is the relaxation time
t
. So the quantity
t
can be derived from the
measurement of the correlation function once l* has been measured, and from
t
the diffusion coefficient D can be calculated. If the particles in the suspension
can be considered as freely diffusing identical spheres, their radius R can be
determined from the Stokes
Einstein equation,
D
¼
kT
ð
2
Þ
6
pZ
R
;
where k is Boltmann's constant, T the temperature, and
Z
the viscosity of the
medium. The factor l* in turn is defined in terms of the scattering properties of
the particles in the suspension, as described by the form factor F(q) and the
structure factor S(q):
16
1
Z
l
/
F
ð
q
Þ
S
ð
q
Þ
q
3
dq
ð
3
Þ
The form factor F(q) is related to the size, shape and refractive index contrast
of the scatterers as used in the scattering of dilute suspensions. The structure
factor S(q) describes the effects of the positional correlations between the
scatterers on the scattering of light. In highly diluted systems, we have S(q)
¼
1
since the particles are far apart and their positions are completely uncorrelated,
and there is no multiple scattering. However, in milk or food emulsions, where
there is maybe a substantial volume fraction (
f
Z 0.1) of dispersed material,
the particles are relatively close to one another. For instance, the average
distance between the surfaces of the casein micelles in milk is about 0.75 of the
particle diameter. Thus, spatial correlations between the scattering particles
become important in the light scattering, and these in turn depend on the
interactions between the particles. If the volume fraction and refractive index
contrast between the particles and the continuous phase are constant
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