# Carbon Allotropes/Epoxy Nanocomposites as Capacitive Energy Storage/Harvesting Systems
## 3 Results and Discussion
### Energy Storage and Harvesting  
#### 3.3.1 AC Measurements  
The ability of a nanocomposite to store electrical energy is expressed by its energy density [22]. The energy density, for all examined systems, was calculated via Equation (1):  
\[\mathrm{U}=\frac{1}{2}\epsilon_{0}\epsilon\mathrm{/E}^{2}, \tag{1}\]  
where \(\epsilon_{0}\) is the dielectric permittivity of the free space, which is equal to \(8.854\times 10^{-12}\) F\(/\)m, \(\epsilon^{\prime}\) is the real part of permittivity and E is the field's intensity. The calculations were made at constant electric field \(\left(\mathrm{E}=1\frac{\mathrm{kV}}{\mathrm{m}}\right)\). In linear dielectric materials, polarization is proportional to the applied electric field, and the energy stored per volume increases with increasing permittivity, at constant applied field. The variation of energy density as a function of frequency and temperature for all nanocomposites with 3 phr filler content is depicted in Figure 5. The illustrated behavior is representative of all examined systems. The energy density increases with temperature as the induced thermal energy facilitates the molecular mobility and the alignment of the dipoles with the externally applied electric field, thus increasing the achieved polarization. At high temperatures and low frequencies, energy density attains high values due to interfacial polarization. Polarization--and therefore also the energy density values--diminish  
Figure 4: Comparative plots of the storage modulus as a function of temperature for the nanocomposites with (a) carbon black; (b) MWCNT; (c) GnP; (d) ND. Insets depict the respective loss modulus as a function of temperature. rapidly with increasing frequency as permanent and induced dipoles do not have sufficient time to align themselves with the rotating external field. At intermediate frequencies and temperatures, a relaxation process arises which is related to the glass to rubber transition of the polymer matrix, also known as \(\alpha\)-relaxation. The relative energy density function, defined by Equation (2), was employed to highlight the contribution of the nanorefinorcements and to estimate the percentage of the stored energy in the nanocomposites compared to the energy stored in the matrix. Its normalized character eliminates the influence of geometric factors. \[\mathrm{U_{relative}}=\frac{\mathrm{U_{composite}}}{\mathrm{U_{resin}}} \tag{2}\]  
Figure 6 showcases the variation of \(\mathrm{U_{relative}}\) as a function of temperature at 160 \({}^{\circ}\)C for all examined systems. In general, the ability of the nanocomposites to store energy increases with filler content due to the higher values of conductivity for the sp\({}^{2}\) hybridized carbon nanoparticles, such as MWCNTs, and dielectric permittivity for the sp\({}^{3}\) hybridized nanodiamonds [16] and the induced heterogeneity of the systems. The nanocomposites with carbon black and MWCNT filler content exhibit enhanced energy storage ability that reaches up to hundreds and thousands of times, respectively, the ability of the neat epoxy matrix at the maximum point of the function at the highest filler concentrations. The enhanced performance of MWCNT and carbon black systems, especially at intermediate and high concentrations, can be attributed to the increased values of conductivity of  
Figure 5: The variation of energy density as a function of frequency and temperature for the nanocomposites with 3 phr: (a) CB; (b) MWCNT; (c) GnP; (d) ND content. these systems. A thorough conductivity analysis of these systems reported in [16] shows that the significant increase of conductivity corresponds to a very narrow variation of the conductive phase content. This increase can be described by means of percolation theory, which predicts that at a critical concentration (also called the percolation threshold), a conductive path is formed, allowing the migration of charge carriers throughout the material [16; 33].