## 3 Results and Discussion
### Activation Energy of Non-Isothermal Cold Crystallization  
The activation energy of the non-isothermal cold crystallization of PLA and its composites was calculated with two different methods, those of [PERSON] and [PERSON]. [PERSON]'s method determines the activation energy for the transport of the macromolecular segments to the growing surface, which is obtained from the maximum conversion rate. The corresponding formula is [53]:  
\[\ln\biggl{(}\frac{\alpha}{T_{p}{}^{2}}\biggr{)}=const-\frac{E_{\alpha}}{RT_{p}} \tag{6}\]  
By plotting the left hand of Equation (6) vs. \(1/T_{\rm p}\), the graph of Figure S5 is obtained. The \(E_{\alpha}\) values of Figure 3 were calculated by the slope after linear fitting of the data (slope = \(-E_{\alpha}/R\)). The Kissinger method assumes that the process occurs as single-step kinetics and does not describe adequately the complex temperature dependence of the crystallization rate [54; 55]. The calculated value of activation energy corresponds to the peak temperature of the crystallization curve and is equal to 107.8 kJ/mol for PLA, in agreement with other studies [56]. All nanocomposites have smaller \(E_{\alpha}\), suggesting that the filler reduced the energy barrier of the non-isothermal cold crystallization of PLA. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Temperature (\({}^{\circ}\)C)** & **100** & **105** & **110** & **115** & **120** & **125** & **130** \\ \hline \(m\) & 2.10 & 2.28 & 1.70 & 1.33 & 1.09 & 0.92 & 0.90 \\ \hline \(K\)(T) & 0.59 & 3.21 & 4.15 & 5.28 & 6.73 & 8.56 & 13.93 \\ \hline R\({}^{2}\)The error caused by the dependence of the extent of conversion on the heating rate when using the Kissinger equation is eliminated by using isoconversional methods [55]. The isoconversional method developed by [PERSON] [57] is the most common differential method used to calculate activation energy of non-isothermal crystallization [58]. The Friedman method allows the calculation of a conversion dependent \(E_{\alpha}\). The \(E_{\alpha}\) values for different \(X_{\text{t}}\) can be calculated by the equation:  
\[\ln\left(\frac{dX}{dt}\right)_{X,i}=Const-\frac{\Delta E_{x}}{RT_{X,i}} \tag{7}\]  
where \(\text{d}X/\text{d}t\) is the instantaneous crystallization rate as a function of time at a given conversion \(X\), \(\Delta E_{X}\) is the effective activation energy at a given conversion \(X\), \(T_{X,i}\) is the set of temperatures related to a given conversion \(X\) at different heating rates \(\alpha_{i}\), and the subscript \(i\) refers to each individual heating rate used. By plotting the left-hand side of the logarithmic form of Equation (7) with respect to \(1/T_{X}\), the graphs of Figure S6 were obtained, with a slope equal to \(-\Delta E_{X}/R\). The effective activation energy is the sum of the activation energies of nucleation and crystal growth. The resulting \(\Delta E_{X}\) for different relative crystallinity is shown in Figure 4a. According to the ICTAC Kinetics Committee recommendations for performing kinetic computations on thermal analysis data, estimating the \(E_{\alpha}\) vs. rate dependence is sufficient for making kinetic descriptions [35]. By calculating the average temperature at each relative crystallinity, the temperature dependence of the effective activation energy was extracted and is presented in Figure 4b. It is clear that \(E_{\alpha}\) depends on the temperature, which implies a complex crystallization mechanism. \(E_{\alpha}\) decreases until \(X_{\text{t}}\) reaches 50%, and then increases until the completion of crystallization. The same effect is noticed with the variation of crystallization temperature. From Figure 4b, it is evident that all nanocomposites crystallized in a lower average temperature range. This inverted bell shaped curve of the cold-crystallization \(\Delta E_{x}\) dependency on relative crystallinity and time has been reported before for PLA [48; 59; 60; 61] and other aliphatic polyesters [63; 64]. [PERSON] et al.