**4. Discussion**

In polymer nanocomposites, traps are potential wells formed by polar groups on molecular chains, which can capture mobile charges and then hinder the motion of mobile charges [17,18]. Some of the electrons or holes may be caught in shallow traps and the extended states, and others may be trapped by the deep traps on molecular chains. Free volumes account for some of the space that is not occupied by atoms in polymers. In free volumes, mobile electrons are accelerated. Electrons captured by traps will lead to local space charge accumulation and then distort the local electric field. However, those that are not captured by traps will keep moving, leading to local current multiplication and Joule heating. Nanodoping can change the trap properties and expansion dynamics of free volumes in nanocomposites [20,21]. Figure 7 summarizes the logical block diagrams of EBEF, EBEG, and EBMD models, illustrating the concept of traps and free volumes, and comparing differences among these three criteria.

After charges are injected into the polymer nanocomposites, some mobile charge carriers are captured by deep traps on molecular chains, resulting in the space charge accumulation and electric field distortion. When the highest electric field exceeds a threshold value, namely *E*(*x,t*)*max* ≥ *EC* [23], DC electric breakdown occurs, which is the criterion of the EBEF model. The accelerated mobile charges in free volumes that are not captured by deep traps will gain energy from the local electric field. When the highest gained electron energy from the constant scale free volume exceeds the deep trap energy, namely [*eλ*0*E*(*x,t*)]max ≥ *uT* [27], DC electric breakdown occurs, which is the criterion of the EBEG model. The property of interfacial region is extremely vital for the distribution of traps in

the bulk of nanocomposites. Accordingly, the electrical breakdown fields calculated by the EBEF and EBEG models change with the increase in nanofiller contents.

**Figure 7.** Schematic diagram of EBEF, EBEG, and EBMD models.

With incorporation of different types of nanofiller, the motion behavior of molecular chains also changes. At a relatively low content of nanoparticles, molecular chains are arranged in an orderly manner in interfacial zones. The mean distance between nanoparticles is smaller than the entanglement tube diameter of the polymer with an increase in nanofiller content, leading to continuous overlapping of Gouy–Chapman layers; then, the nanocomposite system changes from polymer-like to network-like [35,36]. In DC electric breakdown experiments, the electric field is sufficiently strong to force molecular chains to move and rotate if they have a dipole moment. Otherwise, the Coulomb force will act on the molecular chains with occupied deep traps and enlarge the local free volume, leading to larger energy accumulation of accelerated electrons. If the electron energy gain in this expanded free volume is higher than the deep trap energy, namely [*eλfv*(*x,t*)*E*(*x,t*)]max > *uT* [13,25], the electric breakdown may be triggered, which is the criterion of the EBMD model. This model focuses on the molecular chain movement with the deep traps occupied by charges to investigate the influence of charge carrier transport and molecular chain displacement on the DC breakdown strength.

Figure 8 depicts the comparison between the simulated electric breakdown strengths obtained by the three models of EBEF, EBEG, and EBMD and the experimental results of the LDPE/Al2O3 nanodielectrics. It demonstrates that, with the increase in nanofiller content, the breakdown strengths obtained by EBEF, EBEG, and EBMD models all show a trend of

increasing first and then decreasing. The general trends of the simulation and experimental results are similar. However, the simulation results of the EBMD model are in best agreement with the experiments. When the nanofiller content is around 0.5 wt%, the simulation results of EBEG and EBEF deviate greatly from the experimental results. According to the experimental results, the maximum electric breakdown strength is 355.8 kVmm−1, which appears at the nanofiller content of 0.5wt%, while the simulation results of EBEF, EBEG, and EBMD models are 286.3, 312.2, and 356.1 kVmm−1, respectively. It is apparent that the simulation results of the EBMD model are more consistent with the experiments. This indicates that the synergistic effect of deep trap centers in interfacial zones and the tight binding of molecular chains enhance the breakdown performance of LDPE nanocomposites.

**Figure 8.** Comparison of EBEF, EBEG, and EBMD simulation electrical breakdown fields with experimental results.

The larger trap energy and density in nanodielectrics doped with small amounts of nanofillers reduce space charge accumulation and electric field concentration, resulting in the increase in the breakdown strength of nanocomposites. However, the changes in trap energy and density have a limited effect on the breakdown electric field. It is necessary to simultaneously consider the space charge accumulation formed by the trapping of deep traps, the free volume expansion caused by the long-range displacement of the molecular chain driven by the electric field force, and the effect of the energy accumulation of electrons in the expanded free volume on the breakdown strength. Comparative studies show that the energy accumulation of electrons in the expanded free volume due to the long-range displacement of the molecular chain dominates the breakdown strength. When the interaction between the molecular chains in the interface region between the nanofiller and the polymer matrix is enhanced, it is difficult for the molecular chains to undergo long-range displacement under the driving of the electric field force. In this case, the free volume expansion is small and it is difficult for electrons to obtain sufficient energy, so the breakdown strength is increased. Therefore, when designing the structure of the interface region, the interaction between the molecular chains in the interface region can be enhanced by the surface modification method, so that the breakdown strength can be greatly improved. LDPE is a key insulating material for power cables and energy storage dielectric capacitors. Revealing the breakdown mechanism of LDPE nanodielectrics can better develop insulating materials with high breakdown strength. This will provide theoretical and simulation model support for the development of high-performance power cables and energy storage dielectric capacitors.

In addition, the aggregated structure of polymer nanocomposites can be changed to some extent, compared to that of pure polymers. The interface between the crystalline region and the amorphous region, and the interface region between the nanofiller and the polymer matrix, may form different trap distributions. The change in crystallinity and the change in lamellar length may change the interaction strength between molecular chains. Because the molecular chains in the interface region are bound by nanoparticles, the interaction strength between the molecular chains also changes. In future studies, we will correlate aggregated structures with trapping effects and molecular chain interactions through density functional theory [37] and molecular dynamics simulations [38]. Then, the EBMD model will be used to determine the effect of the trapping effect and molecular chain interactions on the breakdown strength. Ultimately, a multiscale model will be established to study the relationship among the aggregated structure, the trapping effect and molecular chain interactions, and the electric breakdown performance.
