*2.1. Charge Injection and Charge Transport in Nanocomposites*

Figure 1 presents a schematic diagram of the charge transport inside a polymer nanocomposite [25]. The one-dimensional coordinate *x* is set in the thickness direction of the nanocomposite, and the thickness of the material is *L*. The left side of the nanocomposite is the cathode and the right side is the anode. Under the action of an applied voltage, the cathode injects electrons into the nanocomposite, and the anode injects holes. Potential barriers exist between the cathode and anode and the nanocomposite interface, which are *uin*(*e*) and *uin*(*h*), respectively. The interfacial barrier hinders the transfer of charges from the electrode into the nanocomposite. Under an externally applied strong electric field, a potential barrier lowering *uSch* appears in the interface barrier, which promotes the transfer of charges to the bulk of the nanocomposite. When the applied electric field is *E*, the Schottky barrier reduction *uSch* is proportional to the square root of the electric field, *uSch* = (*eE*/4π*ε*0*εr*) 1/2. Here, *e* is the electron charge, *ε*<sup>0</sup> is the permittivity of the vacuum in F/m, and *ε<sup>r</sup>* is the dielectric constant of the nanocomposite. We adopt the Schottky thermal emission model to describe the charge injections, *jin*(*e*) and *jin*(*h*), per unit time and unit area of the cathode and anode, into the nanodielectrics [28].

$$j\_{\rm in(\varepsilon)}(0,t) = AT^2 \exp\left(-\mu\_{\rm in(\varepsilon)}/k\_B T\right) \exp\left(\mu\_{\rm Sch}(0,t)/k\_B T\right) \tag{1}$$

$$j\_{\rm in(h)}(L, t) = AT^2 \exp\left(-\mu\_{\rm in(h)} / k\_B T\right) \exp\left(\mu\_{\rm Sch}(L, t) / k\_B T\right) \tag{2}$$

where *A* is the Richardson constant, *T* is the temperature of materials, and *t* is the elapsed time of applied voltage.

**Figure 1.** Schematic of the bipolar charge carrier injection and charge carrier transport in polymer nanocomposites under a ramp voltage.

Under an electric field *E*, there will be a certain concentration of mobile electrons and mobile holes in extended states after the charges in the electrodes are injected into the nanocomposite. Their concentrations are *nM*(*e*) and *nM*(*h*), respectively. These mobile electrons and mobile holes migrate in the extended states driven by the applied electric field. The mobilities of electrons and holes in the extended states are *μ*0(*e*) and *μ*0(*h*), respectively. Due to the existence of polar groups in polymer nanodielectrics, a certain concentration of deep traps is formed. It is assumed that the energy levels of deep traps of electrons and holes are *uT*(*e*) and *uT*(*h*), respectively, and their concentrations are *NT*(*e*) and *NT*(h), respectively. Due to the strong trapping ability of deep traps, mobile electrons and mobile

holes may be captured by them during the migration process. The trapping probabilities of electrons and holes in extended states in the electron deep traps and hole deep traps are *PT*(*e*) and *PT*(h), respectively. After a period of time, the deep trapped electron and deep trapped hole densities are *nT*(*e*) and *nT*(*h*), respectively. In addition, the charges in the deep traps gradually gain energy from the heat bath due to thermal excitation. When the trapped charges gain sufficient energy, trapped electrons and trapped holes can transition to extended states with probabilities of *PD*(*e*) and *PD*(*h*), respectively. The detrapping probabilities *PD*(*e*) and *PD*(*h*) decrease exponentially with the increase in the deep trap levels *uT*(*e*) and *uT*(h). When positive and negative charges meet inside the sample, the recombination between these charges occurs. Mobile electrons may recombine with mobile and trapped holes, and mobile holes may recombine with mobile and trapped electrons. The trapped electrons mainly recombine with the mobile holes, and the trapped holes mainly recombine with the mobile electrons.

The time-dependent change in the charge densities of mobile electrons and mobile holes per unit volume is related to the current density flowing into and out of the control volume. Additionally, trap capturing and recombination lead to a decrease in mobile electrons and mobile holes, while the detrapping of trapped charges results in an increase in mobile electrons and mobile holes. The time-dependent changes in trapped electrons and trapped holes per unit volume are related to charge trapping, charge detrapping, and recombination. Charge detrapping and recombination lead to a decrease in deep trapped electrons and deep trapped holes, while charge trapping leads to an increase in deep trapped electrons and deep trapped holes. Four partial differential equations are needed to describe the dynamic processes of mobile electrons, deep trap electrons, mobile holes, and deep trap holes in nanocomposites, respectively [25,29,30]:

$$\frac{\partial n\_{\rm M(\varepsilon)}}{\partial t} + \frac{\partial \left(n\_{\rm M(\varepsilon)} \mu\_{0(\varepsilon)} E\right)}{\partial x} = -P\_{\rm T(\varepsilon)} n\_{\rm M(\varepsilon)} \left(1 - \frac{n\_{\rm T(\varepsilon)}}{N\_{\rm T(\varepsilon)}}\right) + P\_{\rm D(\varepsilon)} n\_{\rm T(\varepsilon)} - R\_{\rm Mc,\rm Mh} n\_{\rm M(\varepsilon)} n\_{\rm M(h)} - R\_{\rm Mc,\rm Th} n\_{\rm M(\varepsilon)} n\_{\rm T(h)} \tag{3}$$

$$\frac{\partial n\_{T(\varepsilon)}}{\partial t} = P\_{T(\varepsilon)} n\_{M(\varepsilon)} \left( 1 - \frac{n\_{T(\varepsilon)}}{N\_{T(\varepsilon)}} \right) - P\_{D(\varepsilon)} n\_{T(\varepsilon)} - R\_{T\varepsilon, Mh} n\_{T(\varepsilon)} n\_{M(h)} \tag{4}$$

$$\frac{\partial n\_{M(h)}}{\partial t} + \frac{\partial \left(n\_{M(h)}\mu\_{0(h)}E\right)}{\partial x} = -P\_{T(h)}n\_{M(h)}\left(1 - \frac{n\_{T(h)}}{N\_{T(h)}}\right) + P\_{D(h)}n\_{T(h)} - R\_{Mc, \text{Mh}}n\_{M(c)}\mu\_{\text{M}(h)} - R\_{\text{Tr}, \text{Mh}}n\_{T(c)}\mu\_{\text{M}(h)} \tag{5}$$

$$\frac{\partial n\_{T(h)}}{\partial t} = P\_{T(h)} n\_{M(h)} \left( 1 - \frac{n\_{T(h)}}{N\_{T(h)}} \right) - P\_{D(h)} n\_{T(h)} - R\_{Me, Th} n\_{M(c)} n\_{T(h)} \tag{6}$$

The detrapping probabilities *PD*(*e*) and *PD*(*h*) of the deep trapped charges in the nanocomposite are related to the trap energy levels *uT*(*e*) and *uT*(h) as *PD*(*e*,*h*) = *ν*0exp (−*uT*(*e*,*h*)/*kBT*). Here, *ν*<sup>0</sup> is the attempt-to-escape frequency. *RMe,Mh* is the recombination coefficient between mobile electrons and mobile holes. According to the Langevin recombination model, the recombination coefficient *RMe,Mh* is determined by *e*(*μ*0(*e*) + *μ*0(*h*))/*ε*0*ε<sup>r</sup>* [31]. *RMe,Th* and *RTe,Mh* are the recombination coefficient between mobile electrons and trapped holes, and that between trapped electrons and mobile holes, respectively. According to the Shockley–Read–Hall model, the recombination rates *RMe,Th* and *RTe,Mh* are determined by *eμ*0(*e*)/*ε*0*ε<sup>r</sup>* and *eμ*0(*h*)/*ε*0*εr*, respectively [32].

When space charges accumulate inside the nanocomposite, the space charges can distort the electric potential and the electric field. The potential *ϕ* inside the nanocomposite can be calculated by Poisson's equation:

$$
\partial^2 \varphi / \partial \mathfrak{x}^2 = -e \left( n\_{M(h)} + n\_{T(h)} - n\_{M(\varepsilon)} - n\_{T(\varepsilon)} \right) / \varepsilon\_0 \varepsilon\_r \tag{7}
$$

Then, the electric field distribution inside the nanocomposite can be calculated through the negative gradient of the electric potential, namely *E* = −∇*φ*.

#### *2.2. Electrical Breakdown Criteria*

In the EBEF breakdown model, the electric field concentration effect caused by the accumulation of space charges is mainly considered. It is assumed that when the highest electric field *Emax* inside the nanocomposite reaches a certain threshold value *EC*, the material is broken down. The breakdown criterion is *Emax* ≥ *EC* [23].

In the EBEG breakdown model, the process of electron energy gain under the applied electric field after the mobile charge enters the free volume is mainly considered. When the energy gained by the mobile charges is greater than the trapping ability of the deep traps, the nanocomposite is broken down. Assuming that the free volume length is *λ*0, the energy gained by the electron in the free volume is *eλ*0*E*(*x*). The maximum energy of electrons inside the nanocomposite is *eλ*0*Emax*. As shown in Figure 2a, the breakdown criterion of the EBEG model can be obtained as *eλ*0*Emax* ≥ *uT* [27].

**Figure 2.** Electrical breakdown criteria of EBEG (**a**) and EBMD (**b**) models under a ramp voltage.

In the EBMD breakdown model, the charge carrier injection and transport processes inside the nanocomposite under a strong electric field are considered. Furthermore, after the charges are captured by the deep traps on the molecular chain, the electric force acts on the molecular chain to cause its directional displacement. The time-dependent relationship of the displacement *λmol* of the molecular chain is calculated by the following equation [33]:

$$\frac{d\lambda\_{mol}}{dt} = \mu\_{mol}E - \frac{\lambda\_{mol}}{\tau\_{mol}}\tag{8}$$

where *μmol* is the molecular chain mobility and *τmol* is the relaxation time constant of the molecular chain. In addition, *μmol* decreases with the increase in the deep trap energy, *μmol* = *μ*0exp(-*uT*/*kBT*), and *τmol* increases with the increase in the deep trap level, *τmol* = *ν*<sup>0</sup> <sup>−</sup>1exp(*uT*/*kBT*).

The directional movement of the molecular chain will cause the expansion of the free volume. It is assumed that the free volume length is equal to the molecular chain displacement, that is, *λfv*(*t*) = *λmol*(*t*). The mobile charges gain energy in the expanded free volume. If the energy gained by the electrons exceeds the trapping ability of the deep trap, the nanocomposite will be broken down. As shown in Figure 2b, the breakdown criterion of the EBMD model is expressed as [*eλfv*(*t*)*E*(*t*)]max ≥ *uT* [25].

#### **3. Results**

Both the trap energies and the breakdown strengths of LDPE nanodielectrics doped with various fillers first increased and then decreased with the increase in doping content [5,6,11,12]. It is generally believed that the interfacial region is responsible for the excellent electrical properties of polymer nanocomposites. The trap levels in the interfacial region and the interaction between molecular chains are two key factors to improving the breakdown strength of polymer nanocomposites. However, which factor is more influential remains unclear. The main influencing factors of the breakdown strength are determined by comparing the above-mentioned breakdown simulation models with the experimental results. The quantitative relationship between the influencing factors and

the breakdown strength was obtained. We took the LDPE/Al2O3 nanocomposite as an example to compare the three breakdown models. In order to easily compare the simulation results with the experimental results, the parameters in the simulation model were derived from the experimental results [13]. The thickness of the LDPE/Al2O3 nanocomposite was 200 μm and the temperature of the sample was 300 K. We investigated the breakdown properties of pure LDPE and LDPE/Al2O3 nanodielectrics with nanofiller contents of 0.1, 0.5, 2, and 5 wt%. Since the numerical simulation of the charge drift equation should obey the Courant–Friedrich–Levy (CFL) law, the films were discretized into 300 elements and the computational time interval Δ*t* was set as 1 ms [34]. Figure 3 shows the trap density, trap energy level, and attempt-to-escape frequency of pure LDPE and LDPE/Al2O3 nanodielectrics as a function of nanofiller content [13]. *NT*, *uT*, and *ν*<sup>0</sup> were obtained from the experimental results of thermally stimulated depolarization currents, and they all first increased and then decreased with the increase in nanofiller content. Charges in the electrodes may first be transferred to the surface traps in the nanocomposite and then injected into the bulk of the material. In this case, the charge injection barriers can be set to the deep trap levels. The mobilities of mobile electrons and mobile holes in the extended states of the nanocomposite were both set to be 1 × <sup>10</sup>−<sup>13</sup> <sup>m</sup>2V−1s−1. The trapping probabilities of the deep traps were calculated from the trap densities and the carrier mobilities of the mobile charges. The detrapping probabilities of trapped charges were calculated from the trap levels and the attempt-to-escape frequencies. The recombination coefficients of positive and negative charges were calculated from the mobilities of mobile electrons and mobile holes.

**Figure 3.** Trap density (**a**), trap energy (**b**), and attempt-to-escape frequency (**c**) versus filler content of LDPE/Al2O3 nanocomposites.

It can be seen that the trap energy and density are positively correlated with the breakdown strength of the polymer nanocomposites. However, quantitative studies are still lacking about how much the trap energy and density can change the breakdown strength by affecting the space charge accumulation and electric field concentration. In addition, quantitative studies are also lacking about the extent to which the trap energy and density change the breakdown strength by affecting the electric field distortion and the energy accumulation of electrons in a fixed free volume. Moreover, whether the increase in breakdown strength is caused by the binding effect of nanofillers on molecular chains should also be considered. In order to obtain these quantitative relationships, it is necessary to carry out simulation studies of different breakdown models and compare them with the experimental results to determine the primary and secondary influencing factors. Finally, the quantitative relationship between each influencing factor and the variation range in the breakdown strength can be obtained.

## *3.1. Simulation Resutls of EBEF Model*

A voltage having a ramping rate of *kramp* was applied to the electrodes on both sides of the nanocomposite. As the time *t* increases, the voltage *Vappl* applied to the electrodes on both sides of the nanocomposite increases gradually, that is, *Vappl* = *krampt*. The electric field inside the nanocomposite increases gradually, and the charges injected into the material from the electrodes gradually increase. Due to the slow charge transport inside the nanocomposite, the space charges of the same polarity gradually accumulate inside the material. Homogeneous space charges can distort the electric potential and electric field in the nanocomposite. Figure 4 demonstrates the time-dependent changes in space charge distribution and electric field distribution in the pure LDPE and LDPE/Al2O3 nanocomposites. As the voltage increases gradually, the accumulated space charges in all samples increase greatly, and the distortions of the electric fields become stronger and stronger. When the maximum electric field in the material reaches a certain threshold, that is, *Emax* ≥ *EC*, the sample will be broken down. By comparing the experimental results, *EC* was set to 290 kV/mm.

**Figure 4.** Distributions of space charges (**a1**–**a5**) and electric fields (**b1**–**b5**) in pure LDPE and LDPE/Al2O3 nanodielectrics with nanofiller contents of 0.1 wt%, 0.5 wt%, 2.0 wt%, and 5.0 wt%.

With a small amount of doping, the trap density and energy levels of LDPE/Al2O3 nanocomposites increase. The ability of the traps to capture charges is enhanced, and more mobile charges are captured by the traps in the insulating material near the electrodes. The trapped charges of the same polarity near the electrodes increase, so that the electric field near the electrodes is greatly reduced. It can be seen from the Schottky emission equation that the charge carriers injected by the electrodes into the nanocomposite are greatly reduced at low electric fields. This can decrease the degree of electric field distortion inside the nanocomposite. Under the same voltage, the maximum electric field *Emax* in the nanocomposite becomes smaller. This increases the breakdown electric field of the nanocomposite. When a large amount of doping is used, the trap density and energy level of the LDPE/Al2O3 nanocomposite gradually decrease, the space charge distribution becomes longer, and the electric field concentration becomes larger, leading to a larger maximum electric field *Emax* in the nanocomposite. This can lead to a reduction in the breakdown strength of nanocomposites having larger filler contents.

#### *3.2. Simulation Results of EBEG Model*

It is assumed that a certain scale of free volume exists in the LDPE nanocomposites. Charges in the extended states enter the free volumes and are accelerated by the electric field to gain energy. Figure 5 depicts the time-dependent changes in space charge distributions, electric field distributions, and electron energy distributions in pure LDPE and LDPE/Al2O3 nanocomposites. As time passes, the voltage applied to the electrodes on both sides of the materials increases gradually. The space charge accumulation inside the materials increases, the electric field distortion becomes more serious, and the energy gained by the electrons in the free volume also increases. When the charges gain energy beyond the trapping ability of the deep traps, breakdown of the material will occur.

**Figure 5.** Distributions of space charges (**a1**–**a5**), electric fields (**b1**–**b5**), and electron energy gains (**c1**–**c5**) in pure LDPE and LDPE/Al2O3 nanocomposites with nanofiller contents of 0.1 wt%, 0.5 wt%, 2.0 wt%, and 5.0 wt%.

When doped with a small amount of nanofillers, the LDPE/Al2O3 nanocomposites produce more traps having deeper energy levels. More homogeneous space charges accumulate near the electrodes, weakening the electric field distortion inside the nanocomposites. In addition, as the traps become deeper, the trapping ability of the deep traps is increased. Two factors work together to increase the breakdown strength of the nanodielectrics having lower nanofiller weight fractions. After doping a large amount of nanofillers, the trap density and energy levels of LDPE/Al2O3 nanocomposites decrease due to the overlapping of the interfacial regions. At this time, the electric field in the material is seriously distorted, and the trapping ability of the deep traps is weakened. Their combined effects lead to a decrease in the breakdown electric field of the nanocomposites having larger nanofiller contents.
