Chemistry Reference
In-Depth Information
and, the molecular weight distribution is:
M
w
=M
n
¼
1
þ p
approaches 1, the
molecular weight distribution approaches 2. There is experimental confirmation of this. Until now
this discussion was concerned with formation of linear polymers. The presence, however, of
monomers with more that two functional groups results in formation of branched structures. An
example is a preparation of a polyester from a dicarboxylic acid and a glycol, where the reaction
mixture also contains some glycerol. Chain growth in such a polymerization is not restricted to two
directions and the products are much more complex. This can be illustrated further on a trifunctional
molecule condensing with a difunctional one:
It is interesting that this equation tells us that at high conversion, when
p
A
A
A
A
B
A
B
A
+
nA
n
B
B
A
B
B
A
B
B
A
A
A
Further growth, of course, is possible at every unreacted functional group and can lead to gelation.
The onset of gelation can be predicted from a modified form of the Carothers equation [
1
]. This
equation includes an
factor that averages out the functionality of all the
functional groups involved. An example is a reaction mixture of difunctional monomers with some
trifunctional ones added for branching or cross-linking. The average functionality,
average functionality
f
ave
, may be
¼
(2 + 2 + 2 + 3)/4
2.25. The Carothers equation, discussed above, states that
p ¼ðN
o
NÞ=N
o
where
N
o
and
N
represent the quantities of monomer molecules present initially and at a conversion
point
p
. The number of functional groups that have reacted at that point is 2(
N
o
N
). In the modified
equation, the number of molecules that were present initially is
N
o
f
ave
. The equation now becomes:
2
ðN
o
NÞ
N
o
f
ave
p ¼
N
o
/
N
can be replaced by DP and the above expression becomes:
p ¼
2
=f
ave
2
=
DP
f
ave
occurs when the average degree of polymerization becomes
infinite. At that point, the second term in the above equation becomes zero. When that occurs, the
conversion term becomes
It is generally accepted that
gelation
p
c
. It is the
critical reaction conversion
point:
p
c
¼
2
=f
ave
Gelation, however, is less likely to be a major concern in polymerization reactions where only
small quantities of tri- or multifunctional monomers are present. In the preparation of alkyds, for
instance (described further in this chapter), some glycerin, which is trifunctional, is usually present.
