Chemistry Reference
In-Depth Information
ε
is at least 1. The critical value of
q
,
q
cr
, for this condition (which is sometimes
called
gelation
) is then given by
1
q
cr
ð
y
w
2
1
Þ
(9-65)
5
and
1
y
w
2
1
y
w
q
cr
5
(9-66)
1
C
since
y
w
c
:
Since gelation occurs when a fraction 1
1
y
w
of all monomer units is cross-
linked, a completely gelled polymer sample contains one cross-link per weight
average molecule.
If all functional groups in a polymerization are equally reactive and some
monomers have two functional groups, gel may be formed during the polymeriza-
tion reaction. This “gel point” occurs at a degree of conversion to monomer of
polymer,
p
(0
=
p
1) such that
,
,
=ρ
0
y
w
p
1
(9-67)
5
where
ρ
0
is the fra
c
tion of reactive groups that are part of a multifunctional cross-
linking agent and
y
w
is the weight average degree of polymerization that the poly-
mer would have had if all monomers contained one functional group.
Cross-linking can occur in free-radical polymerizations because of chain trans-
fer to polymer (Section 8.8.4) or monomer (Section 8.8.2), as well as the presence
of multifunctional monomers. When it is desirable to retard or suppress cross-
linking, chain transfer agents are added to the polymerization mixture.
Equation
(9-67)
shows why this is helpful.
Consider the copolymerization of bifunctional monomer A with a monomer
BB that contains two functional groups. If all the reactive groups are equally reac-
tive,
Eq. (9-67)
is equivalent to
5
½
A
1
2
½
y
w
5
½
BB
A
1
½
B
p
(9-68)
y
w
2
½
BB
½
B
where the [A] and [B] are concentrations of functional groups and
y
w
is
the weight average degree of polymerization that would be observed in the homo-
polymerization of monomer A under the particular reaction conditions.
F
or exam-
ple, if styrene were being polymerized under conditions such that
y
w
of the
polystyrene being made were 1000, the product would be completely gelled at
full monomer conversion if the reaction mixture contained 0.5
10
2
3
mol frac-
tion of divinyl benzene and the vinyl groups in the latter monomer were equally
as reactive as those in styrene. (Their reactivities are not exactly equal
[23]
.)
Equation (9-67)
indicates that the extent of reaction at which gelation occurs
can be increased by reducing the concentration of divinyl monomer, by reducing
the weight average chain length (increase initiator concentration or add chain
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