Page 120 - Biodegradable Polyesters

98 4 Synthesis, Properties, and Mathematical Modeling of Biodegradable Aliphatic Polyesters

adequate. Considering a simple stoichiometric reaction, the global reaction rate
in terms of the reaction extent, , takes the following form [47]:
] n
[ (G) ∕(S) − (1 − ) n
d [ ] 2n−1 0 0
= k (S) ∕V 0 (4.32)
0
dt (1 − ) 2n−1
with is equal to the cumulative water evaporated up to time t (which is the
experimentally measured quantity) over the total theoretical amount of water that
should be produced and is equal to 0.3. S , G and V are the initial amounts of
0 0, 0
dicarboxylic acid, glycol, and reaction volume and k and n an overall kinetic rate
constant and an overall reaction order.
The theoretical (t) curve can be fitted to the corresponding experi-
mental curve with a great success, even better than the one achieved by
using more complicated kinetic models [43]. The fitting parameters were
3
)
k = 1.73 × 10 −5 −1 ( m mol −1 0.46 for the reaction rate constant and n = 0.73
s
for the reaction rate exponent.
4.4.2.5 Modeling the Effect of Silica Nanoparticles on the Esterification Reaction
Furthermore, modeling of the esterification reaction was attempted in the pres-
ence of silica nanoparticles during the formation of aliphatic polyester nanocom-
posites. From the experimental data, it was found that on increasing the SiO
2
content in esterification, the rate of water production decreases [47]. In addi-
tion, it was clear that the total quantity of water released does not depend on
the nanoparticle concentration. This suggests that the existence of the particles
does not influence the esterification reaction itself. Their main effect is to adsorb
the produced water before it evaporates, altering in this way the water evapora-
tion curve. The simplest model for this phenomenon is to assume very fast water
adsorption/desorption kinetics on the SiO particles. In this case, the evapora-
2
tion kinetics must be explicitly taken into account because it is no more very fast
compared to the other phenomena that occur.
The equation derived to express the variation of the extent of esterification reac-
tion including adsorption/desorption of water on SiO particles takes the follow-
2
ing form [47]:
] n
d (S) [ (G) − (S) + (S) (S) n
0
0
=−k (4.33)
dt V 2n−1
where k and n are an overall kinetic rate constant and reaction order and the reac-
tion volume, V is calculated from
{ ( )}
V = V 1 − 1 − [(S) + (W)] ∕(S) (4.34)
0 0
where the total amount of water (W) in the mixture is the sum of the amounts of
free water (W) and the adsorbed water.
f
In order to estimate it, a sorption/desorption model is employed. Since from
experimental data it was found that water is slowly desorbed from the silica
particles mainly at high reaction times, a complicated model taking into account
the finite rate of water desorption was considered. Accordingly, Ψ is used to