## 3 Results and Discussion
### Dispersion of Functionalized Silica Micro- and Nanoparticles by Ultrasound Micro-Moding Technology
Therefore, the incorporation of nanoparticles had a strong impact on the overall crystallization rate due to differences in crystal growth rate and primary nucleation densities, which became more significant at high and low crystallization temperatures, respectively. Figure 14 also shows that the nucleation density increased exponentially for the two samples at lower crystallization temperatures. These changes in nucleation logically affected the final spherulite size. Thus, a diameter decrease from 120 \(\upmu\)m to 35 \(\upmu\)m and from 80 \(\upmu\)m to 15 \(\upmu\)m was observed for PE99 and PE99-N 3 samples, respectively, when the temperature decreased from 60 \({}^{\circ}\)C to 56 \({}^{\circ}\)C.  
The logarithmic form of the Lauritzen and Hoffman equation [31] was employed to estimate the secondary nucleation constant (\(K_{\rm g}\)):  
\[\ln G+U*/R(T_{\rm c}-T_{\infty})=\ln G_{0}-K_{\rm g}/(T_{\rm c}(\Delta T)f) \tag{3}\]  
where \(G\) is the radial growth rate, \(T_{\rm c}\) is the crystallization temperature, \(T_{\infty}\) is the temperature below molecular motion ceases, \(\Delta T\) is the degree of supercooling, \(f\) is a correction factor calculated as \(2T_{\rm c}/(T_{\rm m}+T_{\rm c})\), \(U*\) is the activation energy, \(G_{0}\) is a constant preexponential factor, \(R\) is the gas constant, and \(K_{\rm g}\) is the secondary nucleation constant. The Lauritzen-Hoffman plot was fitted with straight lines (\(r^{2}>0.97\)) for micro-molded PE99 and PE99-N 3 samples when the \"universal\" values reported by [PERSON] and [PERSON] [32] (_i.e._, \(U*=1500\) cal/mol and \(T_{\infty}=T_{\rm g}-30\) K) and the experimental \(T_{\rm g}\) of PE99 (_i.e._, \(-45\)\({}^{\circ}\)C) were used in the calculation (Figure 15). Figure 14: Change in the nucleation density with isothermal crystallization temperature for PE99 (**garnet**) and PE99-N 3 (**blue**) samples. The nucleation term (deduced from the slope of the plot) mainly influenced the kinetic features at low supercoolings in such a way that crystallization rates could become relatively insensitive to the _U_* and _T\({}_{\infty}\)_ parameters. Therefore, the equilibrium melting temperature was determined since it influenced the degree of supercooling, and consequently the nucleation term. Typical Hoffman-Weeks plots [33] were made with samples crystallized at different temperatures (not shown), leading to equilibrium temperatures of 79.2 \({}^{\circ}\)C and 76.4 \({}^{\circ}\)C for PE99 and PE99-N 3 samples, respectively. The slight change suggests less perfect lamellae upon addition of functionalized silica nanoparticles. The Lauritzen-Hoffman plot allowed for estimating secondary nucleation constants of 1.30 \(\times\) 10\({}^{5}\) and 1.18 \(\times\) 10\({}^{5}\) K\({}^{2}\) for PE99 and PE99-N 3 samples, respectively. These values indicate that the presence of functionalized nanoparticles favored the crystallization process. Thus, enhanced PE99/SiO\({}_{2}\) interfacial interaction may decrease the energy involved in the folding of polyester chains and promote their deposition on existing crystal surfaces during the secondary nucleation process [26,34].