Chemistry Reference
In-Depth Information
Starting with
Eq. (2-30)
instead of
Eq. (2-29)
, it is easy to show that
s
w
=
M
w
5
M
z
M
w
2
1
(2-35)
If M
w
and M
n
of a polymer sample are known, we have information about the
standard deviation s
n
and the variance of the number distribution. There is no
quantitative inf
or
matio
n a
bout the breadth of the weight distribution of the same
sample unless M
z
and M
w
are known. As mentioned earlier, it is often assumed
that the weigh
t a
nd
n
umber distributions will change in a parallel fashion and in
this sense the M
w
=
M
n
ratio is called the breadth of “the” distribution although it
actually reflects the ratio of the variance to the square of the mean of the number
distribution of the polymer (
Eq. 2-34
).
Very highly branched pol
ym
er
s,
like polyethylene made by free-radical, high-
pressure processes, will have M
w
=
M
n
ratios of 20 and more. Most polymers made
by free-radical or coordi
nat
io
n
polymerization of vinyl monomers have ratios of
from 2 to about 10. The M
w
=
M
n
ratios of condensation polymers like nylons and
thermoplastic polyesters tend to be about 2, and this is generally about the nar-
rowest distribution found in commercia
l th
e
rm
oplastics.
A truly monodisperse polymer has M
w
=
M
n
equal to 1.0. Such materials have
not been synthesized to date. The sharpest distributions that have actually been
made
ar
e t
ho
se of polystyrenes from very careful anionic polymerizations. These
have M
w
=
M
n
ratios as low as 1.04. Since the polydispersity index is only 4%
higher than that of a truly monodisperse polymer, these polystyrenes are some-
times assumed to be monodisperse. This assumption is not really justified, despite
the small difference from the theoretical value of unity.
For example,
let
u
s
co
nsider a polymer sample for which M
n
5
100,000,
M
w
5
1.04. In this case s
n
is 20,000 from
Eq
. (2-
33)
.
It can be shown, however
[1]
, that a sample with the given values of M
w
and M
n
could have as much as 44% of its molecules with molecular weights less than
70,000 or greater than 130,000. Similarly, as much as 10 mol% of the sample
could have molecular weights less than 38,000 or greater than 162,000. This poly-
mer actually has a sharp molecular weight distribution compared to ordinary
synthetic polymers, but it is obviously not monodisperse.
It should be understood that the foregoing calculations do not assess the sym-
metry of the distribution. We do not know whether the mole fraction outside the
last size limits mentioned is actually 0.1, bu
t
we kn
ow
that it cannot be greater
than this value with the quoted simultaneous M
n
and M
w
figures. (In fact, the dis-
tribution would have to be quite unusual for the proportions to approach this
boundary value.) We also do not know how this mole fraction is distributed at the
high- and low-molecular-weight
en
ds an
d w
hether these two tails of the distribu-
tion are equally populated. The M
n
and M
w
data available to this point must be
supplemented by higher moments to obtain this information.
We should note also that a significant mole fraction may not necessarily com-
prise a very large proportion of the weight of the polymer. In our last example,
104,000, and M
w
=
M
n
5
