# A Comparison of Electrical Breakdown Models for Polyethylene Nanocomposites
## 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 \(E(x,t)_{max}\geq E_{C}\)[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 \([e\lambda_{0}E(x,t)]_{\max}\geq u_{T}\)[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  
Figure 6: Distributions of space charges (**a1–a5**), electric fields (**b1–b5**), molecular displacements (**c1–c5**), and electron energy gains (**d1–d5**) in pure LDPE and LDPE/Al\({}_{2}\)O\({}_{3}\) nanocomposites with nanofiller contents of 0.1 wt%, 0.5 wt%, 2.0 wt%, and 5.0 wt%. the bulk of nanocomposites. Accordingly, the electrical breakdown fields calculated by the EBEF and EBEG models change with the increase in nanofiller contents. 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 \([e\lambda_{F}(x,t)E(x,t)]_{\text{max}}>u_{T}\)[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/Al\({}_{2}\)O\({}_{3}\) 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  
Figure 7: Schematic diagram of EBEF, EBEG, and EBMD models. increasing first and then decreasing.