High-Performance Carbon Black/Fe₃O₄/Epoxy Nanodielectrics for Electrostatic Energy Storage and Harvesting Solutions

Abstract

The present study explores the energy storage and harvesting properties of nanocomposite systems reinforced with carbon black and magnetite nanoparticles (Fe₃O₄). The systems’ energy storage performance was evaluated under both AC and DC conditions to analyze the impact of temperature, DC charging voltage levels, and varying filler contents on the stored and recovered energy. The experimental findings demonstrated that these systems are capable of efficiently storing and releasing energy on demand via a rapid charge–discharge mechanism. Dynamic mechanical and dielectric analyses revealed significant enhancements in the storage modulus and the energy efficiency of these materials due to the synergistic effects of the nanoparticles and the interactions between them and the polymer matrix. The incorporation of the carbon black and magnetite nanoparticles improves the energy-storage capabilities, supported by augmented interfacial polarization phenomena, which facilitate charge migration and accumulation. These systems exhibit rapid charge and discharge behavior, making them suitable for applications requiring high power density and fast energy storage and recovery cycling. These findings underscore the aptitude of these nanocomposites for high-performance energy-storage solutions, emphasizing their adaptability to applications requiring both high energy density and efficient recovery in tandem with adequate thermomechanical performance.

1. Introduction

The growing demand for high-performance energy-storing materials is driven by the rapid advancement of renewable energy technologies and the need for efficient power management in portable electronic devices [1,2,3]. Traditional dielectric materials often struggle to meet the requirements of high energy density, fast charge/discharge rates, and long-term stability, prompting the search for innovative solutions [4,5,6,7,8,9,10]. Polymer-based dielectric materials have emerged as essential components in advanced energy-storage and electronic systems, owing to their exceptional blend of high flexibility, cost-effectiveness, and customizable functional properties [11,12,13]. These materials are widely used in dielectric capacitors, where their role is pivotal in balancing power fluctuations and ensuring reliable energy storage and release [14,15].

Nanodielectrics have emerged as promising candidates, leveraging the unique electrical and mechanical properties of nanoscale inclusions to address these challenges [16,17]. These materials aim to enhance energy storage, harvesting, and conversion capabilities while supporting diverse applications such as portable and wearable electronics, hybrid electric vehicles, and grid-connected renewable energy sources [18,19]. Within this context, polymer-based dielectric materials, particularly polymer nanocomposites, have gathered considerable interest due to their exceptional ability to store energy at the nanoscale level. The versatility of polymer nanocomposites lies in their tunable properties, which can be optimized by controlling the filler type and content [20,21]. These materials hold immense potential as multifunctional structural energy devices or as components for high-performance electrostatic capacitors [22,23]. Ceramic nanoparticles, when dispersed within polymer matrices, form intricate networks of nanocapacitors, enabling efficient energy storage and rapid charge–discharge cycles. The combination of polymers with high dielectric breakdown strength and ceramic fillers with elevated dielectric permittivity creates composite systems capable of remarkable energy storage and harvesting [24]. This is achieved by leveraging the extended interfacial areas between the polymer and the nanoinclusions, where interfacial polarization plays a dominant role in enhancing dielectric and energy performance for capacitor applications, further underscoring the importance of tailoring microstructures and interfaces [25].

However, challenges such as increased leakage currents with enhanced conductivity values and disparities in fabrication methods must be addressed to unlock their full potential. Methods such as core–shell structures and multilayer designs have proven effective in enhancing dielectric performance, largely by optimizing interfacial properties [26]. Zero-dimensional (0D) fillers, such as spherical conductive or semiconducting nanoparticles, excel in creating uniform electric field distributions, resulting in higher dielectric breakdown strength, reduced dielectric loss, and increased capacitance. These breakthroughs highlight the importance of understanding and engineering fundamental dielectric properties, including polarization mechanisms, filler–matrix interfaces, and hierarchical structures.

Epoxy resin, a popular matrix material in advanced composites, is widely used across industries, including automotive, aerospace, electronics, and energy, due to its exceptional adhesion; compatibility with diverse curing systems; and outstanding thermal, mechanical, electrical, and chemical properties [27]. Carbon black (CB) is known for its excellent electrical conductivity [28,29], while Fe₃O₄ is known for providing extra functionalities to the polymer matrix, such as magnetic properties, contributing superior dielectric properties and augmented interfacial polarization effects [21].

This work aims to explore the thermomechanical, dielectric, and energy-storage properties of hybrid nanodielectrics composed of carbon black and magnetite (Fe₃O₄) nanoparticles embedded in an epoxy matrix. When combined, these nanoparticles exhibit synergistic interactions, significantly enhancing the energy-storage performance of the composite systems and thus demonstrating their capability to meet the demands of modern energy-storage and harvesting technologies. These findings pave the way for the development of next-generation nanodielectrics tailored for sustainable and high-performance applications.

2. Materials and Methods

A series of hybrid polymer nanocomposites was synthesized by varying the filler type and concentration. The matrix utilized was a two-component, low-viscosity epoxy resin system comprising a diglycidyl ether of bisphenol A (DGEBA) as the epoxy prepolymer and an aromatic amine as the curing agent. The commercially available epoxy resin, Epoxol 2004, was sourced from Neotex S.A., Athens, Greece. The reinforcing fillers included carbon black nanoparticles with an average particle size of 13 nm and a specific surface area of 550 m²/g (Plasmachem GmbH, Berlin, Germany) and magnetite (Fe₃O₄) nanoparticles with diameter of less than 100 nm (Sigma Aldrich, Saint Louis, MO, USA).

To prepare the nanocomposites, a predetermined quantity of CB was introduced into the epoxy prepolymer and mixed under ambient conditions using a sonicator for 10 min. Following this, the curing agent was incorporated at a mass ratio of 2:1 (epoxy prepolymer-to-curing agent), and the mixture was sonicated for an additional 5 min. Subsequently, magnetite nanoparticles were added to the formulation, followed by a final sonication step lasting 5 min. The prepared mixture was then transferred into silicon molds and allowed to be cured at room temperature for a week. Post-curing was performed at 100 °C for 4 h to enhance the structural integrity. The filler concentrations were systematically varied, including combinations of 1, 10, and 20 phr (parts per hundred resin per mass) magnetite nanoparticles with 1, 3, and 5 phr CB. Additionally, an unreinforced epoxy specimen was fabricated as a control reference. No surface modification or other chemical treatment was performed before the incorporation of the nanofillers to the polymer matrix.

The morphology, dispersion quality, and particle distribution within the epoxy matrix were evaluated using scanning electron microscopy (SEM) with a Carl Zeiss EVO MA 10 apparatus (Oberkochen, Germany). The thermomechanical performance was assessed via dynamic mechanical analysis (DMA) in the temperature range from room temperature to 100 °C, using a DMA Q800 device (TA Instruments, New Castle, DE, USA) in the three-point bending configuration with a heating rate of 5 °C/min and at the frequency of 1 Hz.

The AC electrical response of the nanocomposites was characterized using broadband dielectric spectroscopy (BDS) within the frequency range of 0.1 Hz to 10⁶ Hz. The measurements were conducted using an Alpha-N Frequency Response Analyzer coupled with a Novotherm temperature control system (Novocontrol Technologies, Montabaur, Germany). The samples were placed in a BDS 1200 dielectric cell in a sandwich capacitor configuration, and frequency scans were performed under isothermal conditions in the range of 30 °C to 160 °C, with temperature increments of 5 °C. The data collection was facilitated by Windeta v5.0 software, and the measurements adhered to the ASTM D150 standards [30].

The energy efficiency under DC conditions was evaluated through charge–discharge tests performed in accordance with ASTM D257 [31] specifications. These tests utilized a High-Resistance Meter (Agilent 4339B, Santa Clara, CA, USA) at room temperature with applied voltages of 50, 100, and 250 V. The specimens were configured as parallel plate capacitors and subjected to a 60 s charging cycle. Further details regarding the charging and discharging sequence can be referenced in [24].

3. Results and Discussion

3.1. Morphology

To assess the distribution and dispersion of the nanoparticles within the polymer matrix, scanning electron microscopy (SEM) images were captured at different magnifications. Representative images for all analyzed systems are presented in Figure 1.

The morphological analysis confirmed the successful fabrication of the nanocomposites. The samples exhibited high quality, with no observable cracks or encapsulated air bubbles within the material. Overall, the nanoparticles were distributed homogeneously throughout the polymer matrix, achieving effective nanodispersion and minimal evidence of severe agglomeration. Statistical image analysis was performed by employing suitable software (Fiji v2.9) [32] to quantify the sizes of the nanoparticles and the level of agglomeration. It can be safely assumed that the range of less than 20 nm corresponds to the carbon black nanoparticles and the range of 20–60 nm illustrates the limited amount of carbon black aggregates. The mean diameter of the carbon nanoparticles was determined to be 16 ± 2 nm. Equivalently, the range of 60–100nm can be ascribed to the magnetite nanoparticles and that of >100 nm primarily to its agglomerates. The mean diameter of the Fe₃O₄ was identified as 97 ± 12 nm.

3.2. Thermomechanical Characterization

The viscoelastic behavior of the investigated nanocomposites was studied via dynamic mechanical analysis in the three-point bending mode. Figure 2 depicts the variation of the storage modulus (E’) and loss modulus (E’’) with temperature for all examined systems. At lower temperatures, the polymer matrix remains in a rigid, glassy state, where the storage modulus attains its maximum value. The addition of nanofillers introduces strong interfacial interactions within the polymer matrix that restrict the molecular mobility and reinforces the structural integrity, leading to greater resistance to deformation under applied stress. Furthermore, they act as barriers that hinder the movement of the polymer chains, limiting their ability to deform elastically. This suppression of mobility increases the rigidity of the material, contributing to a higher storage modulus. Due to their high surface area and small size, the nanofillers create a more uniform dispersion within the matrix, enabling effective stress transfer between the polymer and the filler particles. The storage modulus tends to rise with the incorporation of both types of reinforcing nanoparticles, reaching a maximum of 2.75 GPa in the nanocomposite containing 10 phr carbon black and 20 phr iron oxide—an enhancement exceeding 80% compared to pure epoxy resin. A significant decline in the storage modulus values as the temperature increased marks the polymer matrix’s shift from the glassy state to the elastomeric state, resulting in loss of the load-bearing capacity. These phase transitions are characterized by peak formations in the loss-modulus diagrams.

The variation of the glass transition temperature (T_g) with filler content obtained by the position of the loss-peak maximum offers insights into the interactions taking place within the nanocomposite materials. A significant increase in the T_g values can be observed due to the limitations in the cooperative segmental motion of the polymer chains caused by strong interactions between the nanoparticles and the polymer macromolecules. The higher T_g values exhibited by all hybrid nanocomposites, compared to the neat-epoxy resin, indicate effective adhesion between the reinforcing fillers and the polymer matrix.

3.3. Energy-Storage Performance Under AC Conditions

The energy-storage capability of a nanocomposite is represented by its energy density. For all the investigated systems, the energy density was determined using Equation (1):

$$\mathrm{U}={\displaystyle \frac{1}{2}}{\epsilon }_0{\epsilon }^{\prime }{\mathsf{E}}^2$$

(1)

where ε₀ is the dielectric permittivity of the free space, being equal to 8.854 × 10⁻¹² F/m; ε′ is the real part of the permittivity; and Ε represents the electric field intensity. The calculations were performed under a constant electric field $\left(\mathrm{E}=1{\displaystyle \frac{\mathrm{k}\mathrm{V}}{\mathrm{m}}}\right)$. In linear dielectric materials, polarization is directly proportional to the applied electric field [33]. Consequently, the energy density increases with higher permittivity values when the applied electric field remains constant.

Figure 3 presents the three-dimensional representation of the dependence of the energy density on the frequency and temperature for the nanocomposite with 3 phr carbon black/10 phr Fe₃O₄ content. The variation of U shown in Figure 3 is representative of the behavior of all studied systems. The induced polarization, as expressed by the variation of the dielectric permittivity values, significantly decreases with frequency, since the permanent and induced dipoles, within the material, fail to follow and align with the rapid oscillations of the externally applied electric field vector. In contrast to the frequency, an increase in temperature results in higher mobility of the polymer chains due to the thermal excitation, hence facilitating their orientation with the field and leading to increased values of the real part of the dielectric permittivity and, therefore, the energy density values. The maximum values of U are recorded at high temperatures and low frequencies due to the occurrence of interfacial polarization. The process of interfacial polarization, known also as the Maxwell–Wagner–Sillars effect, is the result of the electrical heterogeneity between the matrix and the inclusions and the accumulation of mobile charges at the interface between the matrix and the fillers. This leads to the formation of large dipoles, which require sufficient time, combined with adequate thermal excitation, to align with the external field. In the region of intermediate temperatures and frequencies, a process is recorded in the form of a “shoulder,” which is attributed to the transition of the polymer matrix from the glassy to the elastomeric phase, also known as α-relaxation. The increase in the induced polarization and energy density observed during the transition is due to the cooperative motion of the segments of the polymer chains, which align more easily with the external field.

Figure 4 illustrates the comparative diagrams of the variation of the energy density as a function of the frequency at 40 °C for all examined systems. The response of the binary systems reveals that in the nanocomposites reinforced with carbon black, the values of the energy density systematically increase with the content of the reinforcing phase across the entire frequency range. The values of the energy density correspond to the variation of the induced polarization with the frequency. The addition of the carbon black nanoparticles enhances the heterogeneity of the systems due to their higher intrinsic conductivity, leading to augmented interfacial polarization. Interfacial polarization is known to occur in the low frequency range, where the large dipoles formed at the interface between the filler and the matrix attain adequate time to align with the externally applied alternating electric field. Therefore, the effect of the carbon black addition is more profound at lower frequencies.

The addition of magnetite nanoparticles leads to notably higher values of U in all systems, which should be attributed to both the interfacial polarization and the nanocapacitor structures formed from the interactions of the carbon black and magnetite nanoparticles. Specifically, MWS polarization plays an important role in achieving high performance as more interfaces are introduced following the Fe₃O₄ addition. A large number of interfaces and nanocapacitors become available as many conductive particles are isolated by the ceramic particles and coated with a thin layer of insulating polymer matrix. Thus, the overall interfacial areas of epoxy/carbon black/Fe₃O₄ available for charge accumulation apparently increase, leading to the rise in the total blocked charges. Secondly, as discussed earlier, the addition of magnetite nanoparticles also provides abundant sites for charge accumulation at the interface between the two fillers. With the addition of the magnetite particles, the proportion of the polymer matrix throughout the system decreases, leading to the reduction in the thickness of the epoxy resin between two adjacent carbon/magnetite particles.

The relative energy density function, determined by Equation (2), was employed to quantify the impact of the nanoreinforcements and assess the proportion of the stored energy in the nanocomposites relative to that in the neat resin matrix. Its normalized nature eliminates the influence of geometric factors, allowing for a more accurate comparison:

$${\mathrm{U}}_{\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}}={\displaystyle \frac{{\mathrm{U}}_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{e}}}{{\mathrm{U}}_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{x}}}}$$

(2)

Figure 5 presents the variation of U_relative with temperature at 0.1 Hz for all the nanocomposite systems under investigation. Generally, the energy-storage capability of the nanocomposite materials improves as the filler content increases. This enhancement is attributed to the higher conductivity values associated with the carbon nanoparticles, the elevated dielectric permittivity of the magnetite nanoparticles, and the induced heterogeneity within the systems. A prominent peak observed at high temperatures is ascribed to interfacial polarization, which arises from the accumulation of free charges at the interface between the inclusions and the polymer matrix. Secondary peaks at lower temperatures reflect the variations in the α-relaxation dynamics between the neat matrix and the nanocomposites.

A closer examination of the U_relative curves reveals the presence of two interfacial polarization peaks, which are influenced by the electrical properties and geometries of the inclusions. Considering the size disparity between the magnetite nanoparticles and the carbon nanostructures—and recognizing that larger dipoles necessitate higher temperatures to align with the field—the peak at the lower temperatures is attributed to the carbon black, while the peak at the higher temperatures is linked mostly to the magnetite nanoparticles. However, the possible presence of nanoparticle clusters interferes with the location of the formed peaks. Considering the curves of U_relative at a constant magnetite content, it becomes apparent that the energy-storage capacity of the nanocomposite systems is significantly enhanced with the increasing carbon black content. This trend is systematic in all curves of Figure 5. However, the amount, or the rate, of increase in the energy-storing ability does not follow a linear relationship with the carbon black content. Moreover, the addition of the magnetite nanoparticles further enhances the dielectric properties and energy-storing ability of the hybrid systems. Taking into account that the effect of the carbon black content upon the energy storage is more pronounced than the corresponding one of the magnetite, the curves in Figure 5c (highest carbon black content and 0, 1, 10 phr content of magnetite) appear to approach at high temperatures because of the prevailing influence of the carbon black. Consequently, the samples containing 20 phr Fe₃O₄ exhibit values 9, 23, and 56 times greater than the pure polymer matrix at the maximum point of the curve for the nanocomposites with 1, 3, and 5 phr carbon black content, respectively.

3.4. Energy-Storage and -Harvesting Performance Under DC Conditions

The calculation of the stored and retrieved energy was performed by integrating the time-dependent current functions, under charging and discharging conditions, via Equation (3):

$$\mathrm{E}={\displaystyle \frac{1}{2}}{\displaystyle \frac{{\mathrm{Q}}^2}{\mathrm{C}}}={\displaystyle \frac{1}{2}}{\displaystyle \frac{{\left(\int \mathrm{I}\left(\mathrm{t}\right)\mathrm{d}\mathrm{t}\right)}^2}{\mathrm{C}}}$$

(3)

where E is the stored energy at the capacitor, Q is the charge, and C is the capacitance of each nanocomposite as evaluated via the BDS measurements at the lowest frequency. More detailed insight into this rationale and this methodology can be found in [34]. Figure 6 contains representative graphs of the stored and recovered energies for the nanocomposite with 3 phr carbon black/10 phr Fe₃O₄ content.

These diagrams confirm the successful process of energy storage and harvesting in the studied systems. All charging curves are above the corresponding discharging curves. The systems quickly reach high values of charging and discharging energy, thus characterizing them as fast-charge/fast-discharge systems. Their ability to promptly store and deliver large energy amounts is desirable in applications requiring high power density as well as the capability for higher charging/discharging currents. The stored and recovered energies rise considerably with the applied voltage, as the applied field lowers the height of the local potential barriers, thereby facilitating charge migration within the nanocomposites in a trapping–detrapping sequence, and also injects more charges.

Figure 7 presents the comparative diagrams of the stored and recovered energies for all the hybrid systems with carbon black under a charging voltage of 250 V. The values of the energy stored in the nanocomposites increase with the carbon black content due to the enhanced electrical properties of the carbon nanoparticles compared to the polymer matrix, as well as the interfacial phenomena caused by the heterogeneity of the systems. The incorporation of the magnetic nanoparticles further boosts the stored energy values, mainly due to the interfacial interactions between the two enclosed phases and the polymer matrix. This results in the grouping of the energy curves, where an increase in the magnetite content leads to an increase in the energy values, though never exceeding the values demonstrated by the next-higher carbon black concentration, with the sole exception being the specimen with 1 phr carbon black/20 phr Fe₃O₄.

The 5 phr carbon black/20 phr Fe₃O₄ nanocomposite recorded almost 5 10⁻⁴ J of stored energy, which, by simply dividing with the volume of the specimen, corresponds to approximately 0.53 mJ/cm⁻³ at room temperature and a 250 V voltage, or else a 80 KV/m applied field. While that may seem reasonably low compared to state-of-the-art nanodielectric and/or commercial capacitors (which approach 10 J/cm⁻³) [18,35,36], the variation of the energy values with the temperature and the applied field must be considered. Based on the findings of the same system depicted in Figure 4, just by applying a field closer to the typical values of the dielectric strength exhibited by DGEBA-based epoxy resins (35–50 kV/mm) [37], the potential energy density, calculated via Equation (1), reaches up to 1.5 J/cm⁻³ at 40 °C, which can be further enhanced at elevated temperatures, as evidenced by the representative performance illustrated in Figure 3. Commercially used biaxially oriented polypropylene (BOPP) at 120 °C exhibits energy density values of 0.25 J/cm³ [38]. Nanodielectrics with a discharged energy density exceeding 10 J/cm³ can now be commonly achieved [39]. While many nanodielectrics have surpassed this benchmark, these high values are typically reported from laboratory-scale experiments using small test areas and short-term performance evaluations, whereas the large-area processing of nanodielectric capacitor films remains rare [40,41].

The relative energies for the charging and discharging procedures are defined, in analogy to Equation (2) for the relative energy density, as the ratios of the stored/retrieved energy of each nanocomposite to the stored/retrieved energy of the neat epoxy at the same initial charging conditions. Notably, the nanocomposites demonstrated exceptional energy storage and recovery performance, surpassing the neat-epoxy matrix by several orders of magnitude.

However, even though the 5 phr carbon black/20 phr Fe₃O₄ specimen exhibited the best performance in the storage process, it trails several systems in the energy recovery process—a behavior that could be attributed to the presence of leakage currents. The incorporation of high concentrations of conductive or semiconductive nanoparticles into polymer nanodielectrics while substantially improving the energy density often results in augmented dielectric losses and more brittle mechanical performance [42,43]. This behavior signifies that the evaluation of the energy-storage performance is not completed without assessing the energy efficiency of the systems, which leads to the introduction of the coefficient of energy efficiency.

The coefficient of energy efficiency (n_eff) is defined as the ratio of recovered to stored energy, with the charging voltage and time serving as key parameters [19]. Figure 8 displays the computed n_eff at identical instantaneous times of t = 5, 10, and 30 s during the charging and discharging processes for hybrid systems reinforced with carbon black and Fe₃O₄. The coefficient of energy efficiency decreases over time across all systems, validating the rapid charging/discharging nature of the systems studied. The hybrid nanocomposites reinforced with carbon black exhibit the highest n_eff values at 100 V rather than at the highest applied voltage. The optimal energy efficiency was recorded for the nanocomposite with 1 phr carbon black/1 phr Fe₃O₄, with a measured coefficient value (n_eff) of 89.6% at 100 V and 5 s. Although the efficiency coefficient value is very high, it should be emphasized that the assessment of a nanocomposite’s ability to store energy and release it during discharge depends not only on the efficiency coefficient but also on the absolute value of the employed energy, which is influenced by the magnitude of the applied field. Currently, nanodielectrics with high discharged energy density are predominantly based on ferroelectric polyvinylidene fluoride (PVDF)-based polymers [44]. However, due to their substantial hysteresis and ferroelectric loss under high-field poling conditions and significantly decreasing energy efficiency at elevated temperatures, these low-temperature polymers may prove unsuitable for practical high-temperature applications [45].

Previous studies identified the percolation threshold of the carbon black/epoxy systems as 5.56 phr [20], while the optimal energy efficiency was exhibited by the nanocomposite with 5 phr carbon black content (n_eff = 80%) [19]. The addition of 10 phr magnetite nanoparticles increased the value of the coefficient to 83.2%, thus maximizing the energy storage without compromising insulation.

This hybridizing by combining different types of nanofillers in order to optimize the dielectric properties and minimize leakage currents is a modern approach to the design of novel polymer nanodielectrics [36]. The incorporation of insulating hexagonal boron nitride nanosheets (BNNS) has often been employed to increase the breakdown strength of polymer nanocomposites and therefore increase the energy density by applying higher electric fields and elevated temperatures [46,47]. However, currently, the scalability of the synthesis process remains a significant challenge because the strong interlayer interactions of 2D nanoplatelets like BNNS severely hinder efficient exfoliation into individual layers, making large-scale production highly difficult [45,48]. Other approaches include the development of multilayered structures and the synthesis of nanoparticles with core–shell morphologies [49,50], although core−shell nanoparticles are difficult to synthesize in large quantities and the hot-pressing methods for multilayer nanocomposites are not promising for large-scale applications, shifting the research focus to the development of multilayer composite films through solution-casting methods [39,45].

As previously mentioned, the calculation of stored and recovered energy was conducted using Equation (3) for both the charging and discharging processes. A limitation of this methodology is the use of the capacitance (C) of each nanocomposite material obtained through the BDS measurements at the lowest frequency (0.1 Hz), assuming that it corresponds to the DC conditions.

Although this convention is necessary for calculating energy during discharging, where measurements are performed in the absence of applied voltage, the energy stored in the system during charging can be calculated according to dielectric theory [33] by incorporating the time-dependent current functions through Equation (4):

$$\mathrm{E}={\displaystyle \frac{1}{2}}{\displaystyle \frac{{\mathrm{Q}}^2}{\mathrm{C}}}={\displaystyle \frac{1}{2}}\mathrm{Q}\mathrm{V}={\displaystyle \frac{1}{2}}\mathrm{V}\int \mathrm{I}\left(\mathrm{t}\right)\mathrm{d}\mathrm{t}$$

(4)

where E is the stored energy in the capacitor, Q the charge, and V the applied voltage under charging conditions.

Figure 9 depicts the stored and harvested energies as a function of time for the nanocomposite, with 1 phr carbon black/20 phr Fe₃O₄ content, as calculated by Equations (3) and (4). The behavior of the systems is representative of all the nanocomposite materials studied. The black curve results from the calculation of the charging energy using the capacitance value of each sample from the dielectric spectroscopy experiments at the lowest frequency. The red curve reflects the corresponding calculation using the applied voltage. Finally, the blue line is the curve of the already-calculated recovered energy.

At first glance, it is evident that calculating the stored energy using the applied voltage brought the storing and recovering curves closer together. It can be concluded, therefore, that conventional calculation using C leads to overestimation of the stored energy values, which increase over time. This observation becomes more evident from the difference in the slopes between the black and red curves. The red curve, representing the energy calculation from Equation (4), as well as the recovering energy curve, shows a plateau over longer times, resembling, to some extent, the charging and discharging process of a nearly ideal capacitor—unlike the black curve, which tends to keep a constant value at a longer period of time.

At this point, it should be noted that while there are indeed differences between the two methods for the calculation of stored energy, the advantages of using Equation (3) for calculation should not be underestimated. The fact that during discharging, measurements are made in the absence of an external field makes the calculation of the corresponding discharging energy via Equation (3) inevitable. Therefore, it seems entirely consistent for the analysis of the energy-storage/recovery capabilities of the nanocomposite systems to be based on the use of the same mathematical expression for energy calculations, hence sharing the same assumptions, limitations, etc. The use of the capacitance value, C, of the nanocomposites measured at even lower frequencies via BDS, while making the entire process considerably more time-consuming, should provide a substantial improvement to the methodology.

4. Conclusions

The present study highlights the successful development and characterization of epoxy-based nanocomposites reinforced with carbon black and magnetite nanoparticles, underscoring their potential for advanced energy-storage applications. Morphological analysis using SEM confirmed the homogeneous distribution and fine dispersion of the fillers within the polymer matrix, further corroborating the quality and integrity of the fabricated systems. The thermomechanical investigation revealed that the addition of both types of reinforcing nanoparticles systematically increases the glass transition temperature and significantly improves the structural integrity. Dynamic mechanical analysis showed that the storage modulus increases significantly with the addition of the nanoparticles due to the stiffness provided by the nanoparticles and their interactions within the polymer matrix. Energy-storage evaluation under AC and DC conditions revealed the strong influence of interfacial polarization in improving the dielectric properties. The systems exhibited increased energy density with filler content, driven by the combined effects of the dielectric permittivity and conductivity provided by the nanoinclusions. The introduction of the magnetite nanoparticles further amplified the energy storage, leveraging the synergistic effects of the multi-filler systems. Notably, the nanocomposites demonstrated exceptional energy storage and recovery performance, surpassing the neat-epoxy matrix by several orders of magnitude. The optimization of the energy efficiency is evident from the calculated coefficients of the energy efficiency. The nanocomposite containing 1 phr carbon black/1 phr Fe₃O₄ achieved optimal energy efficiency, with a coefficient value of 89.6% recorded at a 100 V charging level and at 5 s of the discharging procedure. Overall, these results emphasize the potential of hybrid nanocomposite materials in energy-storage applications, driven by their enhanced electrical properties and rapid energy charge–discharge cycling capabilities. These findings pave the way for further research and development of advanced nanocomposites for high-performance energy-storage solutions.