Chemistry Reference
In-Depth Information
result of rotations about single bonds in the polymer chain and motions of the
segments of the polymer. Nevertheless, the time-averaged effects of the real
solvent-swollen polymer can be taken to be equal to that of equivalent smaller,
impenetrable spherical particles.
For spheres and random coil molecules, the shape factor
ω
in
Eq. (3-60)
is 2.5
and this equation becomes
η=η
0
2
1
5
2
:
5
φ
(3-61)
If all polymer molecules exist in solution as discrete entities, without overlap,
and each solvated molecule has an equivalent volume
V
and molecular weight
M
(the polymer is monodisperse),
φ
then the volume fraction
(phi) of solvent-
swollen polymer coils at a concentration
c
(g cm
2
3
)is
φ5
LcV
=
M
(3-62)
where
L
is Avogadro's number. The two preceding equations yield
5
1
c
η2η
0
η
0
2
:
5
LV
M
(3-63)
In the entity on the left-hand side of
Eq. (3-63)
, the contribution of the poly-
mer solute to the solution viscosity is adjusted for solvent viscosity since the term
in parentheses is the viscosity increase divided by the solvent viscosity. The term
is also divided by
c
to compensate for the effects of polymer concentration, but
this expedient is not effective at finite concentrations where the disturbance of
flow caused by one suspended macromolecule can interact with that from another
solute molecule. The contributions of the individual macromolecules to the vis-
cosity increase will be independent and additive only when the polymer mole-
cules are infinitely far from each other. In other words, the effects of polymer
concentration can only be eliminated experimentally when the solution is very
dilute. Of course, if the system is too dilute,
η 2 η
0
will be indistinguishable from
zero. Therefore, solution viscosities are measured at low but manageable concen-
trations and these data are used to extrapolate the left-hand side of
Eq. (3-63)
to
zero concentration conditions. Then
5
1
c
η2η
0
η
0
2
:
5
LV
M
½
lim
c-
0
lim
c-
0
(3-64)
The term in brackets on the left-hand side of
Eq. (3-64)
is called the
intrinsic
viscosity
or
limiting viscosity number
. It reflects the contribution of the polymeric
solute to the difference between the viscosity of the mixture and that of the sol-
vent. The effects of solvent viscosity and polymer concentration have been
removed, as outlined earlier. It now remains to be seen how the term on the right-
hand side of
Eq. (3-64)
can be related to an average molecular weight of a real
polymer molecule. To do this we first have to express the volume
V
of the equiv-
alent hydrodynamic sphere as a function of
the molecular weight
M
of a
