## 3 Results and Discussion
### Tensile Properties
In this case, the nanocomposites' modulus is described by the following equation:  
\[E_{c}=E_{m}\Bigg{(}\frac{1+\xi\eta V_{f}}{1-\eta V_{f}}\Bigg{)}, \tag{19}\]where \(\zeta=2r\) and \(r\) the aspect ratio of the filler particle and \(\eta\) can be calculated using the following expression:  
\[\eta=\frac{\frac{E_{f}}{E_{m}}-1}{\frac{E_{f}}{E_{m}}+\xi}, \tag{20}\]  
The [PERSON] model is also commonly used on the elastic moduli experimental data of nanocomposites [22]:  
\[V_{f}^{dis}=\Big{(}1-f\Big{(}V_{f}\Big{)}\Big{)}\,V_{f}, \tag{12}\]0  
where \(B\) is a constant parameter. \(B\) is related to the filler geometry and interfacial adhesion between the matrix and the filler particles. When this parameter's value is up to 1, the adhesion between the filler and the matrix is considered poor. If the quality of adhesion is excellent, this value rises higher than 2.5. [PERSON] et al., using [PERSON]'s two-phase model, introduced a three-phase model that includes the matrix, the interfacial region and the filler, assuming that the filler particles (sphere-, cylinders or plate-shaped) are randomly distributed in the matrix [24]. The following equation describes the [PERSON] model:  
\[V_{f}^{dis}=\Big{(}1-f\Big{(}V_{f}\Big{)}\Big{)}\,V_{f}, \tag{12}\]1  
where  
\[V_{f}^{dis}=\Big{(}1-f\Big{(}V_{f}\Big{)}\Big{)}\,V_{f}, \tag{12}\]2  
and  
\[V_{f}^{dis}=\Big{(}1-f\Big{(}V_{f}\Big{)}\Big{)}\,V_{f}, \tag{12}\]3  
where \(h=E_{i}(0)/E_{m}\) is the stiffness ratio of the interfacial modulus around the surface of the filler particle, \(E_{i}(0)\) (GPa), to the one of the matrix \(E_{m}\), \(t_{c}\) (nm) the thickness of the filler's particle and \(\tau\) (nm) is the interphase's thickness. The parameter \(\tau\) equals 0 when the interfacial region's influence is insignificant. In this case, the Ji model drops to the two-phase series-parallel Takayanagi model. A modified Halpin-Tsai equation has been proposed to predict polymer/carbon nanotubes (CNTs) composites' elastic modulus [39]. This modification considers the orientation of the CNTs, the agglomeration state of the filler, and the filler particle's waviness. The composite's elastic modulus follows Equation (19) with the difference of the \(\eta\) factor. According to the modified Halpin-Tsai model, \(\eta\) factor is given by the following equation:  
\[V_{f}^{dis}=\Big{(}1-f\Big{(}V_{f}\Big{)}\Big{)}\,V_{f}, \tag{12}\]4  
where \(f_{R}\) the orientation factor that takes the value \(1/6\) for randomly oriented filler in three dimensions, and the length of the filler is much smaller than the specimen's thickness [40]. \(f_{A}\) is the agglomeration efficiency factor and is given by the Equation (26):  
\[V_{f}^{dis}=\Big{(}1-f\Big{(}V_{f}\Big{)}\Big{)}\,V_{f}, \tag{12}\]5  
where parameters \(\alpha\) and \(\beta\) are related to the agglomeration degree of the filler. The \(f_{W}\) factor represents the waviness of the filler's particles and is given by the following equation:  
\[V_{f}^{dis}=\Big{(}1-f\Big{(}V_{f}\Big{)}\Big{)}\,V_{f}, \tag{12}\]6where \(A\) and \(W\) are the amplitude and half-wavelength of a wavy platelet (in the case of GNPs), respectively. The models described above were applied to the experimental data of SCB-PE/GNPs no BM and w BM nanocomposites. The results of the solely melt-mixed and the ball-milled samples are presented in the supplementary data report in Figure S2a,b, respectively. In both cases of the nanocomposites, the Voight and modified Voight models and the Halpin-Tsai and Ouali's models overestimate the elastic modulus of the composites. On the other hand, both Takayanagi, Ruess and modified Ruess models result in much lower \(E_{c}\) values than the experimental ones for all the prepared nanocomposites. The elastic modulus values derived by the fitting of the Einstein model, while closer to the experimental data for both cases of the nanocomposites, present linear behavior resulting in poor fitting quality.