**5. Conclusions**

The electrical breakdown electric fields simulated by three breakdown models at various nanofiller contents were compared, and the breakdown mechanism of LDPE nanocomposites was illustrated. Doping of Al2O3 nanoparticles into LDPE can change the trap, conductivity, and electrical breakdown properties. The results indicated that the charge trapping, molecular motion, and electron energy gain in free volumes are important factors for the electrical breakdown of polyethylene nanocomposites. Then, space charge transport, electron energy gain, and molecular chain long-distance movement were comprehensively considered to investigate the electrical breakdown mechanism of nanocomposites. EBEF, EBEG, and EBMD models were established via focusing on space charge accumulation due to deep trappings and associated electric field distortion, mobile electrons gaining energy in free volumes, long-distance movement of molecular chains under the Coulomb force expanding free volumes, and accelerating electrons in the enlarged free volumes. Simulation results showed that the EBMD model fits the experimental results much better than the EBEF and EBEG models. The correlation between the long-distance molecular chain movement under the Coulomb force and the electric breakdown characteristics of LDPE was established. The comparisons between simulation results of different models and experiments showed that the electric breakdown electric field of LDPE nanodielectrics is synergistically enhanced by both the strong trapping effect of traps in interfacial regions and the strong binding effect of molecular chains.

**Author Contributions:** Conceptualization, D.M.; Formal analysis, Z.X. and C.Z.; Funding acquisition, D.M.; Resources, Z.X. and Q.W.; Supervision, Q.W. and D.M.; Writing–original draft, Z.X., C.Z., M.H. and Z.G.; Writing–review and editing, M.H., Z.G. and D.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by State Key Laboratory of Advanced Power Transmission Technology (Grant No. GEIRI-SKL-2018-010).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


**Vladimir Bordo \* and Thomas Ebel**

Centre for Industrial Electronics, Department of Mechanical and Electrical Engineering, University of Southern Denmark, Alsion 2, DK-6400 Sønderborg, Denmark; ebel@sdu.dk

**\*** Correspondence: bordo@sdu.dk

**Abstract:** The electrostatic field in a nanocomposite represented by spherical nanoparticles (NPs) embedded into a dielectric between two parallel metallic electrodes is derived from first principles. The NPs are modeled by point dipoles which possess the polarizability of a sphere, and their image potential in the electrodes is found using a dyadic Green's function. The derived field is used to obtain the parameters which characterize the electrical breakdown in a nanocomposite capacitor. It is found, in particular, that for relatively low volume fractions of NPs, the breakdown voltage linearly decreases with the volume fraction, and the slope of this dependence is explicitly found in terms of the dielectric permittivities of the NPs and the dielectric host. The corresponding decrease in the maximum energy density accumulated in the capacitor is also determined. A comparison with the experimental data on the breakdown strength in polymer films doped with BaTiO3 NPs available in the literature reveals a dominant role of the interface polarization at the NP-polymer interface and an existence of a nonferroelectric surface layer in NPs. This research provides a rigorous approach to the electrical breakdown phenomenon and can be used for a proper design of nanocomposite capacitors.

**Keywords:** electrical breakdown; breakdown voltage; nanocomposite; nanocomposite capacitor
