Chemistry Reference
In-Depth Information
9.9
Gel Formation During Copolymerization
and Cross-Linking
[22]
If one of the monomers in a copolymerization is a divinyl compound or any other
entity with a functionality greater than 2, a branched polymer can be formed and
it is possible for the growing branches to interconnect and form infinite molecular
weight products known as
gels
. It is useful to be able to predict the conditions
under which such gel formation will occur. The criteria for this condition are
applicable also to cross-linking of preformed polymers which occurs in radiation-
induced cross-linking, to vulcanization with addition of other reagents, and to
chain-growth and step-growth polymerizations of polyfunctional monomers.
Consider first a sample containing discrete polymer molecules that can be
interconnected either during further polymerization or by a separate reaction on
the macromolecules. Suppose some of the molecules are cross-linked by linkages
formed between randomly selected monomer units. We choose a cross-link at ran-
dom. The probability that the monomer unit on this cross-link resides in a primary
molecule that contains
y
monomer units equals the fraction of all monomer units
that are in
y
-mers. That is, this probability
P
y
is
yN
y
N
y
5
1
P
y
5
yN
y
(9-62)
where
N
y
is the number of
y
-mer molecules in the sample. However, if M
0
is the
mean formula weight of monomeric units in this sample
yN
y
M
0
M
0
N
1
P
y
5
yN
y
5
w
y
(9-63)
where
w
y
is the weight fraction of
y
-mers in the sample.
If a fraction
q
of all monomer units in the sample forms parts of cross-links
and these cross-links are randomly placed, then an additional
q
(
y
2
1) monomer
units in our original
y
-mer are also cross-linked, on the average. That is to say,
the probability that an arbitrarily selected cross-link is attached to a primary chain
that contains
y
monomer units is
w
y
, and it is expected that
q
(
y
1) of these
y
monomer units are also cross-linked. Thus, the initial, randomly chosen cross-link
leads through the primary molecules and other cross-links to
w
y
q
(
y
2
1) additional
primary molecules. Since
y
can have any positive nonzero value, the expected
number of additional cross-links
2
ε
in a molecule that already contains one arbi-
trarily chosen cross-link is
q
N
y
5
1
X
w
y
Þ
5
A
5
w
y
ð
y
1
Þ
5
q
ð
y
w
2
q
ð
y
w
2
1
Þ
(9-64)
2
where
y
w
is the weight average degree of polymerization of the primary chains in
the sample. This molecule can be part of an infinite cross-linked network only if
