Chemistry Reference
In-Depth Information
α
η
is obviously 1, since the polymer is not soluble in media
that are less hospitable than theta solvents. In a good solvent
The lower limit of
α
η
.
1 and increases
M
Δ
, where
with
M
according to
α
η
5λ
λ
and
Δ
are positive and
Δ5
0 under
theta conditions
[7,8]
.Then
3
K
θ
M
ð
0
:
5
1
3
ΔÞ
5
KM
a
½η 5λ
(3-70)
where
K
and
a
are constants for fixed temperature, polymer type, and solvent.
Equation (3-70)
is the Mark
Sakurada (MHS) relation. It appeared
empirically before the underlying theory that has just been summarized.
To this point we have considered the solution properties of a monodisperse
polymer. The MHS relation will also apply to a polydi
sp
erse sample, but
M
in
this equation is now an average value where we denote
M
v
the viscosity average
molecular weight. Thus, in general,
Houwink
KM
v
½η 5
(3-71)
The constants
K
and
a
depend on the polymer type, solvent, and solution tem-
perature. They are determined empirically by methods described
in
Sections 3.3.2
and 3.4.3
. It is useful first, ho
we
v
er,
to establish a definition of
M
v
analogous to
those that were developed for
M
w
, M
n
, and so on in Chapter 2.
3.3.1
Viscosity Average Molecular Weight M
v
We take the Mark
Sakurada equation (
Eq. 3-71
) as given. We assume
also that the same values of
K
and
a
will apply to all species in a polymer mixture
dissolved in a given solvent. Consider a whole polymer to be made up of a series
of
i
monodisperse macromolecules each with concentration (weight/volume)
c
i
and molecular weight
M
i
. From the definition of [
Houwink
η
]in
Eq. (3-64)
,
η
i
=η
0
2
1
5
c
i
½η
i
(3-72)
where
η
i
is the viscosity of a solution of species
i
at the specified concentration,
and [
η
i
] is the intrinsic viscosity of this species in the particular solvent. Recall
that
c
i
5
n
i
M
i
(3-73)
where
n
i
is the concentration in terms of moles/volume. Also,
KM
i
½η
ι
5
(3-71a)
and so
n
i
KM
a1
1
η
i
=η
0
2
1
5
(3-74)
i
