# The Influence of Copper Oxide Particle Size on the Properties of Epoxy Resin
## 3 Results and Discussion
### Mechanical Modeling  
Particle size, particle-matrix adhesion, and particle loading stand out as important determinants in understanding the behavior of a composite material. Several parameters are intricately linked to determine strength and mechanical characteristics such that the quality of particle adhesion to the matrix has a major effect on a composite's strength and toughness. It is observable that when particle sizes are in the micron or greater range, the Young's modulus values of composites are insensitive to their size. On the other hand, as particle sizes approach the nanoscale, a notable increase in stiffness is shown [38]. It was shown that micro-fibrils with sizes between 10 and 30 nm have been seen to influence interphase boundaries more significantly than those in bulk. The composite modulus is expected to increase with decreasing particle size, with a critical particle size of 30 nm [37]. It is generally known that the smaller the dispersed phase, the greater its specific surface area, resulting in shorter distances within the vicinity of the dispersed phase when the volume fraction of the dispersed phase remains the same. This also means that the mechanical responses of a thin layer placed between the stiff dispersed phase and the matrix would differ from those of the dispersed phase. In this study, the effect of particle size on the Young's modulus of polymer composites is shown. To describe this effect, a three-phase Takayanagi model [39] was used which provides better insight into the intricate interaction of various components. The averaged Young modulus of composite \(E_{\text{c}}\) was calculated by following formula:  
\[ \begin{split}\frac{1}{E_{\text{c}}}=\frac{1-\sqrt{\left[\left(r+ \tau\right)/r\right]^{3}}\phi}{E_{\text{m}}}+\frac{\sqrt{\left[\left(r+\tau \right)/r\right]^{3}}\phi}{\left\{1-\sqrt{\left[\left(r+\tau\right)/r\right]^{ 3}}\phi\right\}E_{\text{m}}+\sqrt{\left[\left(r+\tau\right)/r\right]^{3}}\phi (\kappa-1)E_{\text{m}}/\ln\kappa}\\ +\frac{\sqrt{\phi}}{\left\{1-\sqrt{\left[\left(r+\tau\right)/r \right]^{3}}\phi\right\}E_{\text{m}}+\left\{\sqrt{\left[\left(r+\tau\right)/ r\right]^{3}}\phi-\sqrt{\phi}\right\}(\kappa+1)E_{\text{m}}/2+\sqrt{\phi}E_{ \text{r}}}\end{split}  \tag{1}\]  
where \(E_{\text{m}}\), \(E_{\text{r}}\) are, respectively, the [PERSON]'s modulus values of the matrix and reinforcement, \(r\) is the radius of a spherical particle, \(\tau\) is the thickness of the interfacial region, and \(\kappa\) denotes a linear gradient change in the modulus between the matrix and the particle surface, indicating the modulus ratio of the adjacent interface to the particle surface. In Equation (1), the term \(\phi\) denotes the volume fraction of the second phase instead of the mass proportion \(wt\). Thus, this fraction should be calculated using the following equation:  
\[\phi=\frac{1}{1+\frac{\rho_{\text{f}}}{\rho_{\text{m}}}\left(\frac{1}{wt}-1 \right)} \tag{2}\]  
where \(\rho_{\text{f}}\) and \(\rho_{\text{m}}\) represent, respectively, the density of the reinforcing phase and the matrix density. Additionally, the Voigt and Reuss models were used as theoretical frameworks for comparisons with experimental data, aiming to understand the elastic properties of a composite with different particle sizes. The Voigt model [40], based on an assumption of uniform strain, provided an upper bound estimate of the stiffness of the material by averaging the stiffness tensors. \[\overline{E}^{V}=\sum_{j=0}^{k}\phi^{i}E^{i} \tag{3}\]  
where \(\phi^{i}\) is the volume fraction of \(i\)-th phase, labelled \(i=0,\dots,k\). Conversely, the [PERSON] model [41], which assumes uniform stress, provides a lower bound. \[\overline{E}^{R}=\left[\sum_{j=0}^{k}\phi^{i}\left(E^{i}\right)^{-1}\right]^{-1} \tag{4}\]  
The calculations were based on the properties described in this paper with an exception in which 70.5 GPa was taken as the copper oxide stiffness based on the work of [PERSON] et al. [42]. The density of CuO was 6.31 g/cm\({}^{3}\).