Chemistry Reference
In-Depth Information
from Mayo-Lewis equation, for instance)
and r
1
and r
2
are the comonomer reactivity
ratios.
This equation was derived based on
the mechanism widely accepted for olefin
polymerization with coordination catalysts
where chains can propagate by monomer
insertion, terminate through several transfer
mechanism, and LCBs are formed by the
incorporation of vinyl-terminated polymer
chains, commonly called macromonomers.
No other assumption were needed for model
development.
[1,2]
It is interesting to point out that Equa-
tion (1) becomes Stockmayer distribution
for linear chains, that is, for i
¼
0:
[3]
follow Flory's most probable distribution,
Equation (5). MWDs are usually measured
with high-temperature gel permeation chro-
matography (GPC) and expressed in log
scale. Before we can use Equation (5) to
describe the experimental MWD of poly-
olefins, we need to apply two simple math-
ematical transformations. First, we need to
change the CLD into a MWD using the
relation,
wðMWÞ
d
MW ¼ wðrÞ
d
r
(7)
where
is the polymer molecular weight.
Since dMW/dr equals the molar mass of the
repeating unit in the polymer chain (
MW
mw
),
Equation (5) becomes,
2
exp
t
tÞ
wðr; FÞ¼
rt
w
ð
MW
Þ¼
MW
ð
MW
(8)
h
i
q
exp
r
(4)
2
r
ð
F
F
Þ
2
exp
ðrtÞ
where,
2
pb
2
b
mw
¼
1
1
M
n
In addition, if we integrate Equation (4)
over all comonomer compositions we
obtain Flory's most probable chain length
distribution (CLD):
[4,5]
t ¼
mw
¼
(9)
r
n
and
M
n
is the number average molecular
weight of the polymer.
Equation (8) must now be rendered in
log scale through the transformation:
Z
1
w
ð
r
Þ¼
w
ð
r
;
F
Þ
d
ð
F
F
Þ
w
ð
log
MW
Þ
d log
MW
1
2
exp
¼ rt
ðrtÞ
(5)
¼ wðMWÞ
d
MW
(10)
is the
reciprocal of the number average chain
length,
For linear chains, the parameter
t
Consequently:
w
ð
log
MW
Þ
r
n
:
¼
2
:
3026
MW
2
2
exp
ðMWtÞ
t
(11)
rate of transfer
rate of propagation
¼
1
r
n
t ¼
(6)
Figure 1 shows that the MWD of a
polyethylene sample made with two metal-
locene catalysts supported on the same silica
carrier is well represented by the super-
position of two Flory's distributions. Since
we have two single-site catalysts in Figure 1,
the MWD of the combined polymer, W(log
MW
Therefore, in the same way that Stock-
mayer's distribution is the extension of
Flory's distribution to binary copolymers,
Equation (1) is the extension of Stock-
mayer's distribution to non-linear copoly-
mers. We will now start applying Equa-
tion (1), (4), and (5) to several common
polyolefin characterization techniques.
) is described by the weighed sumof two
Flory's distributions,
W
ð
log
MW
Þ
Molecular Weight Distribution of
Linear Chains
¼ m
Zr
wð
log
MWÞ
Zr
þð
1
m
Zr
Þ
w
ð
log
MW
Þ
Hf
(12)
Molecular weight distributions (MWD) of
polyolefins made with single site catalysts
where m
Zr
is the mass fraction of polyethy-
lene produced by the zirconium catalyst.
