## 3 Results and Discussion
### Dynamical Mechanical Analysis
#### 3.3.2 Multifrequency DMTA Analysis  
In order to further investigate the viscoelastic properties of the prepared materials, multifrequency DMTA tests were also carried out. In Figure 8a,b, the storage modulus curves at different frequencies of PLA and PLA-Ar805-8 samples are respectively reported. As expected, the higher the frequency, the higher the storage modulus, and the higher the temperature of damping peak (not shown for brevity), in agreement with previous results obtained on PLLA and PLLA/hydroxyapatite composites [21]. Moreover, the higher the filler content, the higher the storage modulus in the entire range of tested frequencies (0.3-30 Hz). Figure 6: Effective particle volume fraction (\(\phi_{\ell}\), \(\bullet\)), and effective volume per single particle (B) as a function of fumed silica volume fraction (\(\phi_{f}\), \(\Circle\)), according to the Sumita model [68]. Figure 7: Tip penetration in Vicat tests (heating rate 2 \({}^{\circ}\)C/min, load 10 N) on PLA and relative nanocomposites. Moreover, through the application of the frequency-temperature superposition principle, master curves of storage modulus were built, selecting a reference temperature of 30 \({}^{\circ}\)C, in a large frequency interval (about 10\({}^{-20}\)-10\({}^{8}\) Hz), as shown in Figure 9a. Once again, it is evident that funnel silica addition determined a systematic increase of E\({}^{\prime}\) over the whole range of the testing frequencies, especially at elevated filler amounts. From the analysis of the trends of the shift factor (\(log_{10}a_{T}\)) with the temperature reported in Figure 9b, some differences between neat PLA and 2-8 vol% nanofilled samples could be observed, either below or above the \(T_{g}\) zone (about 50-60 \({}^{\circ}\)C). On the other hand, shift factor of 24 vol% nanocomposite results were very similar to that of PLA. Those comparisons appear much more evident considering the activation energy of shift factor \(\Delta E_{sf}\) determined considering an Arrhenius-type dependence, as reported in Equation (4):  
\[loga_{T}=logK-\frac{\Delta E_{sf}}{RT} \tag{4}\]  
where \(K\) is the pre-exponential factor, \(R\) = 8.314 J/molK, and \(T\) the absolute temperature. The slope of the best fit line below (0-40 \({}^{\circ}\)C) and above \(T_{g}\) (55-70 \({}^{\circ}\)C) were considered. The results reported in Figure 9c evidence for composition of 4 vol% a relative maximum of 693 kJ/mol at a low temperature, and a \(\Delta E_{sf}\) value of 476 kJ/mol at a high temperature. These findings seem to be related to the good dispersion of filler, which progressively decreases with silica content, and consequently affects the polymer mobility in both glassy and rubbery states. From multifrequency analysis, the dependence of \(T_{g}\) on frequency \(F\) could be also considered. The correspondent Equation (5) was utilized for the determination of activation energy at \(T_{g}\) (\(\Delta E_{\rm Tg}\)), as an indicator of the chain mobility in the transition from the glassy to the rubbery phase. \[logF=logF_{0}-\frac{\Delta E_{Tg}}{R\ T_{g}} \tag{5}\]  
where \(F_{0}\) is the pre-exponential factor, \(R\) = 8.314 J/molK, and \(T_{g}\) is the temperature of the damping peak at the selected frequency \(F\). The results of best fit line are reported in Table 5, and show a higher activation energy for PLA (623 kJ/mol) with respect to FS nanocomposites, in analogy with the previous findings for PLLA and hydroxyapatite filled composites up to a concentration of 50 wt% [21]. It is also worth noting that \(\Delta E_{\rm Tg}\) of PLA corresponds to \(\Delta E_{\rm sf}\) (621 kJ/mol) at high temperature, whereas the values of nanocomposites result in between the correspondent \(\Delta E_{\rm sf}\) values shown in Figure 9c. Figure 8: Storage modulus in multifrequency DMTA analysis of (a) PLA and (b) PLA-Ar805-8 nanocomposite at various frequencies in the range 0.3–30 Hz (heating rate of 0.5 \({}^{\circ}\)C/min). Figure 9: Multifrequency DMTA analysis on PLA and relative nanocomposites.