## 3 Results and Discussion
### Dispersion of Functionalized Silica Micro- and Nanoparticles by Ultrasound Micro-Moding Technology
Plots of \(\log(-\ln(1-\chi(t-t_{0})))\) against \(\log(t-t_{0})\) (Figure 11) allowed for determining the indicated crystallization parameters, which are summarized in Table 3. Avrami exponents for the neat polyester remain in a narrow range (_i.e._, 2.60-2.94) and have an average value of 2.78. The determined Avrami exponent indicates a predetermined (heterogeneous) nucleation and a spherical growth under geometric constraints. Note that a slight deviation is observed with respect to the theoretical value of 3 and that a value close to 4 should be expected for a sporadic (heterogeneous) and homogeneous nucleations. It should also be pointed out than homogeneous nucleation usually requires high undercooling, which does not correspond with the performed experiments. The exponent slightly decreased for the nanocomposite (_i.e._, 2.54-2.14, with 2.33 being the average value), suggesting an increase in geometric constraints upon incorporation of the well-dispersed nanospheres. Exponents determined for both samples were found to vary without a well-defined trend within the selected narrow temperature interval. Reciprocal crystallization half-times (1/\(\tau_{1/2}\)) are also summarized in Table 3. This parameter is directly determined from DSC isotherms (_i.e._, it corresponds to the inverse of the difference between thr crystallization start time and half-crystallization time) and can be useful to test the accuracy of the Avrami parameters since an estimated value can be obtained from them (_i.e._, 1/\(\tau_{1/2}=(Z/\ln 2)^{1/n}\)). Figure 11: Avrami plots of isothermal crystallization of PE99 (**a**) and PE99-N 3 (**b**) at the indicated temperatures. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Sample** & \(\mathbf{T\left({}^{\circ}\text{C}\right)}\) & \(\mathbf{Z\times 10^{6}\left({}^{\text{s}-^{\text{n}}}\right)}\) & \(\mathbf{n}\) & \(\mathbf{k\times 10^{3}\left({}^{\text{s}-^{\text{1}}}\right)}\) & \(\mathbf{1/r_{10^{3}}\times 10^{3}\left({}^{\text{s}-^{\text{1}}}\right)}\) & \(\mathbf{(Z/\text{ln2})^{1n}\times 10^{3}\left({}^{\text{s}-^{\text{1}}}\right)}\) \\ \hline \multirow{4}{*}{PE99} & 56 & 18.48 & 2.60 & 15.23 & 17.93 & 17.5 \\  & 57 & 0.28 & 2.94 & 5.89 & 6.82 & 6.67 \\  & 58 & 0.077 & 2.81 & 2.92 & 3.31 & 3.32 \\  & 59 & 0.0028 & 2.79 & 0.86 & 0.97 & 0.99 \\ \hline \multirow{4}{*}{PE99-N 3} & 56 & 98.56 & 2.42 & 21.98 & 25.97 & 25.58 \\  & 57 & 42.03 & 2.31 & 12.84 & 15.12 & 15.05 \\ \cline{1-1}  & 58 & 2.69 & 2.54 & 6.37 & 7.62 & 7.36 \\ \cline{1-1}  & 59 & 1.89 & 2.14 & 2.12 & 2.54 & 2.52 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Main crystallization kinetic parameters determined by DSC for the neat polyester and its nanocomposite with 3 wt. % of functionalized silica nanospheres. Figure 10: (**a**) Exothermic DSC peaks of isothermal crystallizations of PE99 (**garnet**) and PE99-N 3 (**blue**) samples at temperatures between 56 and 59 \({}^{\circ}\)C; (**b**) development of the relative degree of crystallinity of PE99 (**garnet**) and PE99-N 3 (**blue**) samples at different crystallization temperatures. Variation of the overall rate constant with temperature for the neat polymer and its nanocomposites was also evaluated (Table 3). The rate for the nanocomposite increased (_i.e._, from 2.12 \(\times\) 10\({}^{-3}\) s\({}^{-1}\) to 21.98 \(\times\) 10\({}^{-3}\) s\({}^{-1}\)) when crystallization temperature decreased (_i.e._, from 59 \({}^{\circ}\)C to 56 \({}^{\circ}\)C), a trend that was also logically observed for PE99. More interestingly, the nanocomposite showed a remarkably higher crystallization rate than the neat polyester at all test crystallization temperatures (e.g., 21.98 \(\times\) 10\({}^{-3}\) s\({}^{-1}\) and 15.23 \(\times\) 10\({}^{-3}\) s\({}^{-1}\) were determined at 56 \({}^{\circ}\)C for PE99-N 3 and PE99, respectively). Therefore, incorporation of functionalized silica nanoparticles had a significant influence on the crystallization process and, logically, on the final material properties.