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236
Kc
R
θ
=
1
+
16
π
2
R
G
2
3
λ
2
sin
2
θ
2
⎛
⎝⎜
⎞
⎠⎟ +
...
⎛
⎝⎜
⎞
⎠⎟
1
M
w
+
2
A
2
c
+
...
⎛
⎝⎜
⎞
⎠⎟
The equation looks complicated, but is easily managed by a computer
program or a worksheet.
In a light scattering experiment the scattering intensity is measured for several
concentrations (including pure solvent), several angles (simultaneously or
sequentially, depending on the type of instrument). As a result, we obtain a
data matrix from which we can obtain – at the same time – M
w
, A
2
and R
G
.
Thus, light scattering is an extremely powerful method for characterizing
macromolecules. It is used a lot by scientists both in industry and academia.
You may convince yourself by a quick literature search.
6.2.5.
Light
scattering provides M
w
and R
G,z
in
case of
polydispersity
Note that we introduced M
w
for M, indicating that for
polydisperse
systems
we
always obtain the weight-average molecular weight
. In fact, this can be
proved easily. Assume a situation at very low angle (P(
θ
) = 1) and very low
concentration (2A
2
c = 0), where we have a mixture of different sizes (indexed
i). The Rayleigh factor is additive, i.e. it is the sum of the Rayleigh factors from
each group of molecules, so that the equation simplifies to:
R
θ
=
R
θ
,
i
i
n
∑
=
Kc
i
M
i
=
K c
i
M
i
=
i
n
∑
i
n
∑
K
c
i
M
i
i
n
∑
c
i
i
n
∑
c
i
i
n
∑
=
KM
w
c
⇓
Kc
R
θ
=
1
M
w
In polydisperse systems also the radius of gyration becomes an
average value. Interestingly, it becomes another type of average
(proof omitted here), namely the z-average defined by:
R
G
2
=
R
G
2
z
=
N
i
M
i
2
R
G
,
i
2
i
=
1
n
∑
N
i
M
i
2
i
=
1
n
∑
NOTE!
In light scattering
the molecular
weight M of
polydisperse
samples is always
M
w
– the weight
average molecular
weight.
In contrast, the
radius of gyration
is z-average.