Page 114 - Biodegradable Polyesters

92 4 Synthesis, Properties, and Mathematical Modeling of Biodegradable Aliphatic Polyesters

Kinetic Rate Constants The kinetic rate constants usually depend on temperature,
as well as catalyst type and concentration [48]. In literature [49, 50], it has been
proposed that the esterification and polycondensation reactions are acid catalyzed
and that the corresponding rate constants can be expressed as

k = k C ; i = 1 − 5 (4.30)
i i,0 acid
2⋅
where k i,0 are the «true» rate constants in l mol −2⋅ min −1 and C acid denotes the
concentration of acid groups defined as the sum of the concentration of carboxylic
end groups (tSA) and the carboxylic groups of the free acid (2SA), that is,
C = {2 (SA) + (tSA)} ∕V (4.31)
acid
The assumption whether kinetic rate constants used in the esterification of the
poly(alkylene succinates) are indeed acid catalyzed is investigated in a following
section.

Integration of the System of Differential Equations and Estimation of Kinetic
Parameters The system of differential Equations 4.19–4.26, together with
Equations 4.12–4.18, was integrated by applying the Runge–Kutta fourth order
method with varying step size. In order to have the mole number of every com-
ponent in the reaction mixture as a function of time, the kinetic rate constants
have to be evaluated [38, 39, 51, 52]. From a detailed analysis [43], it was finally
proposed that the number of parameters that need to be evaluated is only two,
namely k and k .
1
6
Simulation Model Results Initially, the assumption was tested that succinic acid
can act as its own catalyst in the esterification reaction. In Figure 4.6, the experi-
mental results on the esterification of PPSu are compared to the theoretical model
predictions using kinetic rate constant that are either acid catalyzed (dashed and
dotted lines) or not (solid line). As can be seen, the simulation of the experimental
data by the theoretical model is very good when the kinetic rate constants used
are not acid catalyzed. However, when the kinetic rate constants are assumed to be
acid catalyzed, using Equations 4.30 and 4.31, the experimental data are not pre-
dicted equally well. Using values to accurately predict the initial rate data, the final
data are underestimated. In contrast, when such values are used to predict the final
experimental data, the initial data are overestimated. Thus, it was concluded that
in the synthesis of the poly(alkylene succinates) studied here, the presence of the
metal catalyst tetrabutoxy titanium (TBT) leads to a poor activity of self-catalyzed
acid. This was also observed for PBSu by Park et al. [42]. Therefore, Equations 4.30
and 4.31 were not used and only parameters k and k need to be estimated. The
1 6
values of these parameters were calculated for every different system studied from
fitting to the experimental data. The final values are reported in Table 4.2. Notice
that these values are correct only for the specific catalyst type.
The effect of the catalyst molar ratio on the rate of the PPSu esterification reac-
tion is examined next. Figure 4.7 shows the variation of extent of the esterification
reaction measured from the amount of water collected versus time, at several