**1. Introduction**

Power cables are the main equipment in urban transmission grids and offshore wind power transmission [1–6]. Direct current (DC) power cables have the advantages of long transmission distance, large transmission capacity, and low power loss, and they are the key electrical equipment for large-scale reception of new energy power generation. Under the action of DC voltage, the power cable has no capacitive current and can realize longdistance power transmission. The aging of the insulating material of power cables under the DC electric field is slow, and its lifespan is greatly prolonged [1,3–6]. Moreover, the breakdown electric field of the insulating material under DC voltage is 2–3 times higher than that under AC voltage [7], which improves the safety margin of the DC power cable. Low-density polyethylene (LDPE) is the main insulating material of power cables, and its electrical insulation performance is important for the safe and reliable operation of power cables [1–6]. Polymer nanocomposites (PNCs) have excellent properties, such as lower electrical conductivity, higher breakdown electric field, less space charges, higher thermal

**Citation:** Xing, Z.; Zhang, C.; Han, M.; Gao, Z.; Wu, Q.; Min, D. A Comparison of Electrical Breakdown Models for Polyethylene Nanocomposites. *Appl. Sci.* **2022**, *12*, 6157. https://doi.org/10.3390/ app12126157

Academic Editors: Ioannis F. Gonos, Eleftheria C. Pyrgioti and Diaa-Eldin A. Mansour

Received: 8 April 2022 Accepted: 14 June 2022 Published: 17 June 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

stability, and higher mechanical strength [5,6,8–12]. PNCs, known as third-generation insulating materials, have broad application prospects. The experimental results demonstrate that doping a relatively low content of nanoscale fillers in polyethylene can form deep traps, leading to the reduction in the electrical conductivity of the nanocomposites and the improvement in the breakdown strength [5,6,9,11–14]. Advanced LDPE nanocomposites can be used as insulating materials for DC power cables [5,6,12] and energy storage capacitors [15,16], improving the capacity of power cables to transmit electrical energy and the energy storage density of capacitors. Power cables and energy storage capacitors are key equipment for the centralized transmission of large-scale offshore wind power, providing support for the supply of clean energy to cities.

It is generally believed that the excellent electrical properties of polymer nanocomposites originate from the interfacial region between the nanoparticles and the polymer matrix [6,12]. The multi-core interfacial region model proposed by Tanaka et al. [17,18] and the multi-region structure model proposed by Li et al. [19] show that deep traps are formed in nanocomposites when a small amount of doping is used. Deep traps near the electrodes trap more charges and reduce the number of charges injected. This can suppress the space charge accumulation, reduce the electric field concentration, and improve the breakdown electric field. By comparison, traps with higher energies can reduce the effective charge carrier mobility, reduce electrical conductivity and Joule heating, and improve breakdown performance. The interfacial region models proposed by Nelson et al. [20] and Min et al. [21] show that the interfacial regions not only form deep traps, but also constrain the motion of molecular chains. Under the action of the Coulomb force, the molecular chains undergo directional displacement, which affects the size of the surrounding free volumes. This, in turn, changes the breakdown performance of nanodielectrics. The interfacial region models show that increasing the trap level and/or the interaction between molecular chains in the interfacial regions can improve the breakdown strength of polymer nanocomposites.

Experiments and simulations indicate that the breakdown of polymer materials under a strong electric field is related to physical processes such as electric field distortion and electron energy gain. Tanaka et al. [22] used pulsed electroacoustic equipment to test the space charge distributions and the electric field distributions of LDPE under the action of a strong electric field. It was found that positive space charge packets are formed in LDPE when the electric field is higher than a threshold value. As the positive space charge packets move toward the cathode, the electric field in front of the charge packets gradually increases, and the material is broken down when the maximum distorted electric field in LDPE reaches the breakdown electric field. Chen et al. [23] considered the formation and migration of space charges in LDPE, and established a polymer breakdown model based on the accumulation of space charges and the corresponding electric field distortion. The relationship between the DC breakdown electric field of LDPE and the thickness of samples was calculated. It was found that the breakdown electric field has an inverse power function relationship with the thickness. Choi et al. [24] also used a breakdown model based on the electric field distorted by the space charges to calculate the breakdown characteristics of multilayer polymers with partial barrier contact, and found that partial barrier contact between multilayer structures enhanced the breakdown strength of multilayer dielectrics. From the viewpoint that the electric field force acts on the trapped charges and affects the molecular chain motion, and to comprehensively consider the charge transport, the long-range motion of molecular chains, and the electron energy accumulation, we established a charge trapping and molecular displacement breakdown (CTMD) model for polymer nanocomposites [13,25]. The energy accumulation process of electrons in the free volume expanded by the long-range motion of the molecular chain was simulated, and the relation between the breakdown electric field of the polyethylene nanocomposites with the nanofiller content, the applied pressure, the thickness of the sample, and the ramping rate was obtained. The results are consistent with the results of the electric breakdown experiments.

The above analysis shows that the strong trapping effect of traps and the strong interaction between molecular chains in the interface regions are the two key factors to improve the breakdown strength. However, which of the trap trapping effect and the molecular chain interaction is more influential is still unclear. To clarify the factors of the breakdown characteristics of polyethylene nanocomposites having the greatest influence, this study compared three breakdown models, namely, the electric field distortion, the electron energy gain in a fixed-scale free volume, and the energy gain of electrons in an expanded free volume caused by the motion of molecular chains. By comparing the simulation results with the experiments, the electric breakdown mechanism of polyethylene nanocomposites was clarified. In the present work, we determined the quantitative roles of trapping effects and molecular chain interactions on breakdown strength. This paper provides simulation methods and data support for the improvement in the breakdown strength of polymer nanocomposites.
