## 3 Phase Content and Dispersion of Nano-Inclusions
The quantification of the \(\gamma\) phase was more difficult, as exclusive FTIR bands corresponding to the \(\gamma\) phase appeared as shoulders (see Figure 2). [PERSON] et al. [14; 49] analyzed FTIR absorptions by assuming the Lambert-Beer law and calculated the absorption coefficients, K\(\alpha\) and K\(\beta\), at the respective wave numbers of 766 and 840 cm\({}^{-1}\). In this manner, according  
Figure 2: Infrared absorption bands typical of the \(\beta\) and \(\alpha\) phases were identified, and they are presented separately for neat PVDF films (PVDF-20 and PVDF-80) and hybrid ones with non-functionalized nanoparticles (NF-PVDF) and functionalized nanoparticles (F-PVDF). to [14, 49], the relative fraction of the \(\beta\)-phase content in a sample containing mainly \(\alpha\) and \(\beta\) phases is:  
\[F(\beta)=\frac{A_{\beta}}{\left(K_{\beta}/K_{\alpha}\right)A_{\alpha}+A_{\beta}}= \frac{A_{\beta}}{1.26A_{\alpha}+A_{\beta}} \tag{1}\]  
where F(\(\beta\)) represents the \(\beta\)-phase content; K\(\alpha\) and K\(\beta\) are the absorption coefficients with respective values of 6.1 \(\times\) 10\({}^{4}\) and 7.7 \(\times\) 10\({}^{4}\) cm\({}^{2}\)mol\({}^{-1}\); A\(\alpha\) and A\(\beta\) are the absorbances at 766 and 840 cm\({}^{-1}\) calculated using the incident and transmitted intensities of the IR waves. \[A_{n}=\log\frac{I_{n}^{0}}{I}n=\alpha,\beta \tag{2}\]  
This expression has been widely used in the literature [13, 14, 18, 19, 20, 21, 22, 48], and in our case, it gives the following values for the \(\beta\)-phase content in the four samples (see Figure 3):  
IR observations support the conclusion that both neat PVDF films and hybrid ones presented a similar content of electroactive phase with a slight increase (as expected) in the presence of the nano-inclusions. As reported before [42], we decided to use as nano-fillers the isotropic nanoparticles of Ni\({}_{0.5}\)Zn\({}_{0.5}\)Fe\({}_{2}\)O\({}_{4}\) (NZFO-NPs). Previous work has already demonstrated that the interaction between the negatively charged surface of the NZFO-NPs and the positively charged polymer CH\({}_{2}\) groups promotes nucleation of the \(\beta\) phase in PVDF [19]. Remarkably, when hydrophilic molecules functionalize the surface of the nano-fillers, a uniform dispersion of the nano-fillers is obtained. These hydrophilic parts interact with the C-F polar bonds in the PVDF matrix through hydrogen bonding, and all dipoles along the molecular chains of the polymer can align. Consequently, the \(\beta\)-phase content is enhanced [30, 31] as well as the polymer structure. In order not to destroy the flexible properties of the PVDF, we worked at a very low loading mass percentage (\(<\)1% wt) [42] with very small nanoparticles of average diameter of 3.5 nm (see Figure 4a-c in the Supplementary Materials for more details on the nanoparticles' morphological and magnetic characterizations). It is worth noticing here that ferromagnetic nanoparticles behave like strong nano-magnets; thus, as soon as they are sufficiently nearby (closer than 10 nm) they attract each other. Once they agglomerate, it is very difficult to disperse them. The four samples were studied using SEM images of both backscattered and secondary electrons, BSEs and SEs, respectively. Figure 4a presents the SEM image obtained for the neat PVDF film (PVDF-20) in which no contrast can be seen for both the BSE and SE images. In Figure 4b,c, we compare non-functionalized NPs embedded in the PVDF film (NF-PVDF) with their functionalized NPs (F-PVDF) counterparts. The BSE images provide a clean view of the dispersion of the nanoparticles, as they are related to the Z  
Figure 3: \(\beta\)-phase content for neat PVDF films (PVDF-20 and PVDF-80) and hybrid ones with non-functionalized nanoparticles (NF-PVDF) and functionalized nanoparticles (F-PVDF). number of the material imaged. We found that functionalization was a key parameter for dispersing the magnetic nanoparticles in the film.