Page 123 - Biodegradable Polyesters

4.4 Mathematical Modeling of the Synthesis of Aliphatic Polyesters 101

Herein, the subscript x stands for the number of methylene units used and takes
the values x = 2, 3, 4 for the ethylene, propylene, or butylene ester, respectively
The forward reactions are facilitated by the by-product removal either by flow
of an inert gas or by maintaining reduced pressure, or a combination of the two.
The overall reaction rate is influenced by a combination of the intrinsic reaction
kinetics, diffusion of the reactive end groups, change of polymer degree of crystal-
lization, and diffusional limitations on account of desorbing volatile by-products
(i.e., glycol and water) [54–56]. Several models have been developed in literature
describing the rate of change of the concentration of the species present during
polycondensation of PET as a function of time and distance from the interface
[54–59]. Having two independent variables, partial differential equations must
be set and solved, including a number of kinetic, diffusional, and crystallization
parameters [54–56]. Furthermore, the results of these models are used to fit only
a few experimental data points. Therefore, in order to keep the modeling as sim-
ple as possible and not use more adjustable parameters than the experimental data
points, simple modeling approaches are usually followed [57–59]. This approach
was originally developed for the solid-state polycondensation (SSP) of PET and
successfully applied by our group in modeling the SSP of PET with activated car-
bon black nanoparticles [60].

4.4.3.2 Development of the Mathematical Model
The rate of change of hydroxyl and carboxyl end groups is described by the fol-
lowing expressions [59]. More details can be found in Ref. [45]:
d[OH]
t =−2k [OH] − k [COOH] [OH] (4.39)
2
dt 1 t 2 t t
d[COOH] t =−k [COOH] [OH]
dt 2 t t (4.40)
where [OH] and [COOH] denote the actual “true” hydroxyl and carboxyl end-
t
t
group concentration, respectively.
According to Ma and Agarwal [57, 59], the rapid slowdown in SSP kinetics at
high [ ] values can be represented by the transesterification and esterification
reactions only when accounting for a part of the carboxyl ([COOH]) and hydroxyl
end groups ([OH]) to be rendered temporarily inactive, [OH] ,[COOH] .Then,
i
i
the actual concentration of OH and COOH in Equations 4.39 and 4.40 are
expressed as [OH] = [OH] − [OH] and [COOH] = [COOH] − [COOH] . i
i
t
t
The apparent inability of a fraction of OH and COOH groups to participate
in the polycondensation reactions can be due to chemical degradation leading to
unreactive chain ends (such as vinyl end groups), side reactions, or to the reactive
ends being unable to approach each other owing to diffusional limitations [57, 61].
Such limited extent of mobility of some chain ends could be a result of their being
restricted by relatively short chain segments linking them to crystalline parts or a
result of their having been incorporated in crystalline parts as defects. Duh [62]
also accounted for such inactive groups in his modeling approach.