Page 109 - Biodegradable Polyesters

4.4 Mathematical Modeling of the Synthesis of Aliphatic Polyesters 87

if we define as r the ratio of the initial mole fraction of the diacid over the diol
(r < 1), then the average degree of polymerization is defined as
DP = 1 + r (4.4)
n
1 + r − 2rp
From Equations 4.3 and 4.4, it follows that at p = 95% conversion, when r = 1,
the degree of polymerization is 20; when r is lowered to 0.9 or 0.5, the correspond-
ing values become 10 and 2.7, respectively. Hence, an excess of diol leads to much
lower degree of polymerizations.
In order to achieve high-molecular mass products in reasonable reaction
times, small amounts of externally added strong acids (such as sulfuric acid or
p-toluenesulfonic acid) were employed as catalysts. Under these conditions,
[HA] in Equation 4.1 is the concentration of the catalyst. Making the same
assumptions as previously, the dependence of the reaction extent and average
degree of polymerization on reaction time are written as
1 ′ ′
− 1 = N k t ⇒ DP = 1 + N k t (4.5)
n
0
0
1 − p
The polyesterification becomes a much more economically feasible reaction
when it is catalyzed by an external acid. The self-catalyzed polymerization is not
a useful reaction from the practical viewpoint of producing high polymers in
reasonable reaction times.
4.4.2
Mathematical Modeling of the Esterification Reaction for the Synthesis of Aliphatic
Polyesters

4.4.2.1 Literature Survey
As was reported in the previous section, the theoretical modeling of the polyesteri-
fication reaction has been well known for about 50 years since the pioneering work
of Flory [34]. Furthermore, the kinetics of synthesis of aromatic or alipharomatic
polyesters, such as PET, has been extensively studied because of its widespread
production worldwide [38, 39]. There are also a few publications dealing with the
kinetics of PPT [7] and PBT [40, 41]. In contrast, the models published in liter-
ature predicting the esterification of the production of aliphatic polyesters are
very limited. Park et al. [42] in 1998 were the first who developed a model for the
synthesis of PBSu. Flory’s self-catalyzed reaction model was used for uncatalyzed
reactions, together with a new complex model suitable for catalyzed esterifica-
tion. The ability of the diacid used to act as a self-catalyst was also addressed there
[42]. Subsequently, Bikiaris and Achilias [43–45] develop a theoretical mathe-
matical model to predict the kinetics of the esterification and polycondensation
reaction of three poly(alkylene succinates) (PESu, PPSu, and PBSu). The func-
tional group approach was followed. Furthermore, the kinetics of the tetrabutyl
titanate-catalyzed homogeneous succinic acid/1,4-butylenediol (BD) and hetero-
geneous terephthalic acid/BD esterification was investigated by Hu et al. 2010
[46]. A kinetic model considering two main esterification reactions as well as one