Page 108 - Biodegradable Polyesters

86 4 Synthesis, Properties, and Mathematical Modeling of Biodegradable Aliphatic Polyesters

The rate of a step polymerization is, therefore, the sum of the rates of reaction
between molecules of various sizes. The kinetics of such a situation with innumer-
able separate reactions would normally be difficult to analyze. However, kinetic
analysis is greatly simplified if one assumes that the reactivities of both functional
groups of a bifunctional monomer (e.g., both hydroxyls of a diol) are the same,
the reactivity of one functional group of a bifunctional reactant is the same irre-
spective of whether the other functional group has reacted, and the reactivity of a
functional group is independent of the size of the molecule to which it is attached.
These simplifying assumptions, often referred to as the conceptofequal reactivity
of functional groups, make the kinetics of step polymerization identical to those
for the analogous small molecule reaction [35].
The rate of a step polymerization is conveniently expressed in terms of the con-
centrations of the reacting functional groups. In the case of polyester formation,
two such groups appear, namely, carboxyl, –COOH, and hydroxyl, –OH. The rate
of polymerization can then be expressed as the rate of disappearance of carboxyl
groups −d[–COOH]/dt given by
−d[−COOH] = k[−COOH][−OH][HA]
dt (4.1)
Two quite different kinetic situations arise from Equation 4.1 depending on the
identity of HA, that is, on whether a strong acid is added as an external catalyst
providing two different modeling approaches, that of self- and external-catalyzed
polymerization.
In the absence of an externally added strong acid, the diacid monomer acts as
its own catalyst for the esterification reaction. For this case, [HA] is replaced by
[–COOH] in Equation 4.1. Assuming that the two functional groups are nearly
stoichiometric, and setting [–COOH] = N, the final equations expressing the vari-
ation of reaction extent, p and polymer number average degree of polymerization
(NADP), DP with time can be written as [34, 35]
n
1 2
(1 − p) 2 = 2N kt + 1 (4.2)
0

DP = 1 = √ 1 + 2N kt (4.3)
2
n 0
1 − p
where, N is the initial concentration of hydroxyl or carboxyl groups and k a
0
kinetic rate constant.
Since the reaction time and degree of polymerization appear as the first and sec-
ond powers, respectively, the polymer molecular weight will increase very slowly
with reaction time except in the early stages of the reaction. This means that very
long reaction times are needed to obtain a high-molecular-weight polymer prod-
uct. The production of high polymers requires reaction times that are too long
from the practical viewpoint [35].
It should be stressed here that Equation 4.3 is only valid when stoichiometric
amounts of the two monomers are initially present. When the diol is in excess and