## 3 Results and Discussion
### Activation Energy of Non-Isothermal Cold Crystallization
suggested that this deviation from the expected reduction of \(E_{\alpha}\) with the increase of \(T\), which is not consistent with the Hoffman-Lauritzen theory, may be a sign of secondary crystallization [63; 65]. All nanocomposites show smaller \(E_{\alpha}\) than neat PLA. The curves of the nanocomposites of Figure 4 shift to lower values almost parallel to the curve of PLA, suggesting their crystallization mechanism is comparable [42]. Similar curves and \(E_{\alpha}\) values have been reported for PLA in the literature [48]. All methods  
Figure 3: Activation energy of the non-isothermal cold crystallization of PLA and its nanocomposites calculated by the [PERSON] method. show the same trend in the reduction of \(E_{\alpha}\), which follows the order: PLA > PLA Ag > PLA TiO2 > PLA ZnO, regardless of the temperature and the conversion. The method of [PERSON] can be used to study the \(E_{\alpha}\)-T dependence [66]. The resulting values were used for the calculation of the Hoffman-Lauritzen parameters by fitting the equation developed by [PERSON] and [PERSON] onto the data [66; 67]:  
\[E_{\alpha}=U^{*}\frac{T^{2}}{(T-T_{\infty})^{2}}+K_{\mathrm{g}}R\frac{T_{m}^{0 \,2}-T^{2}-T_{m}^{0}T}{\left(T_{m}^{0}-T\right)^{2}T} \tag{8}\]  
where \(U^{*}\) is the activation energy of the segmental jump, \(T_{\infty}=T_{\mathrm{g}}-30\) K, \(T\) is the average temperature at a given relative crystallinity, \(T_{m}^{0}\) is the equilibrium melting temperature (calculated by the linear Hoffman-Weeks method at 443 K, similar to values reported in the literature [68; 69; 70]), and \(K_{\mathrm{g}}\) is the nucleation constant. In contrast with the models derived from the Avrami equation, such as the Ozawa and Mo models, which ignore secondary crystallization and calculate exponents with questionable physical meaning, the \(U^{*}\) and \(K_{\mathrm{g}}\) are parameters that are physically meaningful in the context of polymer crystallization. Figure 4: Dependence of the Friedman effective activation energy on (**a**) relative crystallinity and (**b**) the average temperature for PLA and its nanocomposites. The analysis would be more complete with additional data from non-isothermal melt crystallization, but the type of PLA used crystallizes extremely slowly from the melt. The Levenberg-Marquardt method was used as a non-linear fitting algorithm (from the built-in software of Origin Pro 9, from OriginLab Corporation), using only data points corresponding to X < 60% to increase the fit quality [64; 71]. In that temperature range, PLA crystallizes according to Regime III, a high-rate nucleation stage where the mean separation of nuclei approaches the width of molecular stems [72]. The resulting values for \(K_{\text{g}}\) and \(U^{\ast}\) appear in Table 2.  
The \(U^{\ast}\) values are of the same order as the universal value (6.27 kJ/mol). The \(U^{\ast}\) values of PLA nanocomposites were lower than those of PLA, suggesting that the transportation of PLA chains in the presence of the fillers is faster [59]. Both \(U^{\ast}\) and \(K_{\text{g}}\) values of neat PLA are in good agreement with those reported in the literature [59]. It was found that \(K_{\text{g}}\) values decrease with the incorporation of various fillers, decreasing the energy barrier of the nucleation process [73; 74]. This result is consistent with the analysis of crystallization half times, and the measured effective activation energy, as described above.