# Space Charge Accumulation and Decay in Dielectric Materials with Dual Discrete Traps
## 4 Results and Discussion
### Time Dependent Free and Trapped Charge Densities at the Same Position  
Figure 6a-c demonstrates the free and trapped charge densities as a function of time at the same position \(x=2\Delta x\) in the dielectric material subjected to the voltage of 6.0 kV, 7.5 kV, and 9.0 kV, respectively. The quasi-free charges in shallow traps injected from electrodes will be captured by trap level 1 and trap level 2 simultaneously. Since the trapping probability of trap level 1 is larger than that of trap level 2, the increasing rate of trap level 1 is larger than that in trap level 2. Accordingly, significantly more charges are accumulated in trap level 1. In the meantime, the quasi-free charge density decreases with time. In addition, the charge migration and the decrease of charge injection  
Figure 5: Average net space charge densities (**a**) and external current densities (**b**) as a function of time in the dielectric material containing two discrete trap levels subjected to various voltages. also cause the reduction of quasi-free charges. The reduction of quasi-free charge density transfers the charge trapping/de-trapping dynamics in the case of constant quasi-free charge density to that in the case of constant total charge density. This leads to the decrease of charge density in trap level 1. The charges in trap level 1 will be transferred to trap level 2 via the transport states gradually. Finally, the charges in the material will reach thermodynamic equilibrium. At thermodynamic equilibrium, the charge densities in trap level 1 are larger than that in trap level 2 both in Figure 6.  
The simulation results of charge trapping/de-trapping dynamics in dielectric materials having two discrete trap levels imply that the poling duration of applied DC voltage can influence the energy distribution properties of accumulated space charges. For short poling duration, more quasi-free charges may be captured by the trap level 1 (relatively shallower trap centers). Whereas, the trapped charges in trap level 1 will transfer to trap level 2 (relatively deeper trap centers) via the transport states for a long poling duration. The differences in the energy distribution of accumulated space charges will lead to the differences in space charge decay properties as shown in Figure 7. In the simulation model, we apply a voltage to the sample for some time (\(t_{c}\)) and then the sample is short-circuited to simulate the space charge decay properties. When the charging time \(t_{c}\) is \(1\times 10^{2}\) s, a large portion of charges are trapped in trap level 1, so the decay rate of charges after the sample is short-circuited is much higher than that with longer charging time. The same trends in the decay rate of charges were observed under the application of different voltages. After trap level 2 is fully filled, the decay rate of charges at a relatively longer time becomes the same. The charge decay properties of LDPE were investigated experimentally to demonstrate the charge trapping/de-trapping results obtained theoretically and numerically. LDPE samples with a thickness of 100 \(\upmu\)m were charged by corona discharging for 120 s, 300 s, and 600 s at room temperature and at the humidity of 40%. The samples were charged to around \(-5\) kV and their surface potentials were then measured by a high-voltage electrostatic voltmeter (P0865, Trek, Lockport, NY, USA) with a non-contact probe (3453 ST, Trek, Lockport, NY, USA). Figure 8a shows how the surface potential decay properties change with charging time. As the charging time increases, the decay rate of surface potential decreases, because more charges are transferred into deep traps. From the surface potential decay curves, we calculated the charge densities and trap energies of LDPE. There are two discrete traps with the trap energies of 0.93 eV and 1.0 eV in the sample as shown in Figure 8b. The charge density de-trapped from shallow traps with the energy of 0.93 eV decreases with charge time, while that from deep traps increases.