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104
θ
-conditions (θ-solvent or
θ
-temperature) refers to conditions where weak
attractive forces exactly cancel the excluded volume effect, and α = 1 and
independent of the chain length. When we wish to determine C
∞
experiments
should by conducted at θ-conditions. Alternatively, α should be known from
additional investigations.
How are
θ
-conditions found for a given polymer? Interestingly, the excluded
volume effect also influences thermodynamic properties of the system,
allowing the use of e.g. osmometry or light scattering to find
θ
-conditions. This
is discussed further in Section 3.
Inserting the equation α
∝
n
0-0.
1 in the general equation we see that:
θ
-solvent:
r
2
∝
n
⇔
r
=
r
2 0.5
∝
n
0.5
Good solvent:
r
2
∝
n
1.2
⇔
r
=
r
2 0.5
∝
n
0.6
Or, taking the square roots and combining:
r
=
r
2 0.5
∝
n
0.5
−
0.6
Note that the basic equation for a random coil is still the same, and that R
G
still obeys the relation: R
G
∝
M
a
. If we plot log R
G
as a function of log M should
for a random coil obtain a straight line with slope 0.5 in a θ-solvent and 0.6 in
a good solvent. Compare to the corresponding values for compact spheres
and rods, respectively.
2.2.9. How
to determine C
∞
from experiments?
The classic way to determine C
∞
consists of
analysing the radius of gyration (R
G
) (in a
θ
-
solvent, or knowing
α
2
) for different molecular
weights (of the same polymer). Samples with
different molecular weights (M) (a homologous
series) may be obtained by controlled
degradation of long chains or by special
fractionation methods.
Once we know M
i
(from e.g. light scattering) we
Figure 46. Homologous series:
Same chemical composition,
different chain lengths