Chemistry Reference
In-Depth Information
The following paragraphs include sample calculations that illustrate the practi-
cal application of the Carothers equation to step-growth polymerizations which
yield branched polymers.
Consider the simple alkyd recipe shown in
Table 7.1
, part (i). Alkyds are
polyesters produced from polyhydric alcohols and polybasic and monobasic acids.
They are used primarily in surface coatings. The ingredients of these polymers
contain polyfunctional monomers and it is possible that such polymerizations
could produce a thermoset material during the actual alkyd synthesis. This is of
course an unwanted outcome, and calculations based on the Carothers equation
can be used to adjust the polymerization recipe to produce a finite molecular
weight polymer in good yield. The recipe can also be adjusted to provide other
desirable characteristics of the product, such as an absence of free acid groups
that may react adversely with some pigments.
In the example of
Table 7.1
, part (i), there are 4.2 equivalents of acid groups
and 4.4 equivalents of alcohol. Therefore, from
Eq. (7-16)
, f
av
5
2(4.2)/(4.4)
5
1.91. Then, with
Eq. (7-19)
at p
5
0.9, X
n
5
2/[2
2
(1.91)(0.9)]
5
7.2 and at
p
22. There are four monomers that can conceivably be
incorporated into this polymer and it is clearly impossible to specify a regular
repeating
gr
oup, as was done in Chapter 1 for linear polymers.
Since X
n
is finite at p
5
1, the recipe listed in
Table 7.1
, part (i), will not pro-
duce cros
s-
linked polymer (i.e., it will not “gel”). Note in this connection that the
limit of X
n
for polymerization of bifunctional monomers is infinite at p
5
5
1.0, X
n
5
2/(2
2
1.91)
5
1
(
Eq. 7-20
). This simply means that the whole reaction mixture would be reduced
to a single molecule if all the functional groups could actually be reacted.
The simple recipe we have been discussing is modified in
Table 7.1
, part (ii),
by increasing the relative concentrations of the phthalic anhydride and glycerol.
The coreactive functional groups are now in balanc
e
and f
av
5
2.10.
From
Eq. (7-19)
, when the reaction mixture has gelled X
n
is infinite and thus p at
the gel point is given by the equality: 2
8.8/4.2
5
pf
av
5
0, from which the limiting value
2
of p
0.95. According to this model, the reaction cannot be taken past 95% con-
version of functional groups without producing an intractable, insoluble product.
Note that substitution of p
5
2 yields a negative
value for X
n
. The Carothers equation obviously does not hold beyond the degree
of conversion at which X
n
is infinite.
It is pertinent to ask at this point how well the foregoing predictions work. In
fact, there is a basic error in the Carothers model because an infinite network
forms when there is one cross-link p
er
weight avera
ge
molecule (Section 7.9).
Th
at
is
to say, gelation occurs when X
w
rather than X
n
becomes infinite. Since
X
w
$
1 into
Eq. (7-19)
with f
av
.
5
X
n
(Section 2.7), reaction mixtures should theoretically “gel” at conversions
lower than those predicted by the Carothers equation.
A method to calculate the molecular weight distribution (and X
w
) during the
course of an equilibrium step-growth polymerization is described in
Section 7.4.3
.
Such calculations are of great value in understanding the course of such reactions
but they are not generally any more effective than the simpler Carothers equation
