## 4 Influence of Ageing on Charge Transport Properties and Discussion  
### Influence of Ageing on Trap Distributions  
Figure 6 illustrates how charges detrap from the sample surface and migrate along the self-built electric field. To explain the variation of the decaying rate of silicone rubber samples with different ageing times, trap distributions are analyzed below. It is agreed that the surface trap density related to the surface potential decay rate can be depicted by the following equation [15]:  
\[N_{\mathrm{ST}}(E_{\mathrm{ST}})=\frac{\varepsilon_{0}\varepsilon_{\mathrm{r}} }{edl_{\mathrm{s}}}t\frac{\mathrm{d}\phi_{\mathrm{s}}(t)}{\mathrm{d}t}, \tag{1}\]  
where \(E_{\mathrm{ST}}\) is the surface trap energy in eV, \(N_{\mathrm{ST}}\) is the density of surface traps in m\({}^{-3}\) eV\({}^{-1}\), \(\varepsilon_{0}\) is the permittivity of vacuum in Fm\({}^{-1}\), \(\varepsilon_{\mathrm{r}}\) is the dielectric constant of samples, \(e\) is the elementary charge in C, \(d\) is the thickness of the sample in m, \(l_{\mathrm{s}}\) is the thickness of surface traps located in insulating material of about 1 \(\mu\)m, \(t\) is the decay time of the surface potential in s, and \(\phi_{\mathrm{s}}(t)\) is the surface potential in V.  
Figure 5: Volume conductivities measured at the applied voltage of 5 kV and 20 \({}^{\circ}\)C of silicone rubber samples before and after ageing by ac voltage with a harmonic voltage component for different times. Figure 6: Schematic of the detrapping of electrons (**a**) and holes (**b**) from surface traps and migration in the bulk of silicone rubber. Prior to calculating the trap distribution, fitting curves of the potential decay results need to be obtained. Firstly, \(\varphi\) (\(t\)) can be simplified as the function [16]:  
\[\phi_{\text{s}}(t)=\phi_{\text{s0}}+A_{1}\exp(-t/\tau_{1})+A_{2}\exp(-t/\tau_{2}), \tag{2}\]  
where \(\varphi_{\text{s0}}\) is the initial surface potential of insulating materials charged by corona in V, \(A_{1}\) and \(A_{2}\) are two undetermined coefficients in V; and \(\tau_{1}\) and \(\tau_{2}\) are two undetermined relaxation times in s.  
According to the charge detrapping theory proposed by [PERSON] [17], trap energy \(E_{\text{ST}}\) is proportional to the retention time of charge carriers in traps, and it can be obtained by the equation:  
\[E_{\text{ST}}=k_{\text{B}}Tln(v_{\text{ATE}}t), \tag{3}\]  
where \(k_{\text{B}}\) is the Boltzmann constant, \(T\) is the absolute temperature of silicone rubber during surface potential decay experiments in K, and \(v_{\text{ATE}}\) is the attempt-to-escape frequency in s\({}^{-1}\). Additionally, the attempt-to-escape frequency can be expressed as \(k_{\text{B}}T/h\), where \(h\) is the Planck constant. Surface charge detrapping theory was used to analyze the surface potential decay characteristics of the silicone rubbers, and the hole trap and electron trap distribution characteristics were calculated as shown in Figure 7. Figure 7a shows the relationship between the distribution of hole traps in silicone rubber and the aging time. The surface traps of the pristine silicone rubber are very deep. The maximum trap density corresponds to a trap energy level of 0.87 eV and the corresponding trap density is \(1.67\times 10^{22}\) m\({}^{-3}\)eV\({}^{-1}\). The hole trap energy of the silicone rubber decreases with the aging time, and the longer the aging time is, the shallower the trap will be. The maximum trap density of the silicone rubber aged for 1 and 8 months corresponds to trap energy levels of 0.80 and 0.75 eV, respectively. The relationship between the distribution of the electron trap with the aging time is similar to that of the hole trap. Shallow electron traps gradually increase with aging time as shown in Figure 7b. The change of the trap distribution is related to the molecular group and molecular chain structure of the material under the influence of power frequency superimposed harmonic voltage [18].