## 7 Numerical Results
### Post-Insulator in SF\({}_{6}\)  
In this section, the models previously described are compared to establish the surface charge accumulation phenomenon along the gas-solid interface of a post-insulator embedded in a gas chamber filled with SF\({}_{6}\). The 2D axisymmetric geometry of the model, also considered in [31; 38; 39], is illustrated in Figure 4. The solid insulator of radius \(r_{I}=48\) mm and height \(h_{I}=140\) mm is fed from the top by a high-voltage (HV) electrode, while the chamber of radius \(r_{O}=180\) mm and height \(h_{O}=284\) mm is the low-voltage (LV) electrode. The SF\({}_{6}\) filling the chamber has a pressure of \(P=0.5\) MPa absolute, and the corresponding transport parameters to be used in the DDR model are listed in Table 1. The temperature is considered fixed at \(T=293.15\) K in the subsequent simulations. The solid insulator is made of alumina-filled epoxy resin with \(\varepsilon_{r,I}=5\), \(\sigma_{I}=3.33\cdot 10^{-18}\) S/m, and \(\sigma_{S}=0\) S.  
The voltage applied to the HV electrode is increased from 0 V to 15 kV with a rise time of \(10^{-3}\) s, while the LV electrode is referred to as ground potential (0 V). The simulation runs for \(t=8000\) h, by using a backward differentiation formula (BDF) or maximum order 2. The trend of \(\varrho_{S}(t)\), evaluated with the _td_s approach, is illustrated in Figure 5. For low external ionization, as is the case, \(\sigma_{G}\) is much less than \(\sigma_{I}\), and a comparison with the EQS formulation with constant conductivity is made by using \(\sigma_{G}=1.23\cdot 10^{-17}\) S/m (corresponding to \(\kappa=4.5\cdot 10^{-19}\)). Using quadratic basis functions for \(\varphi\) and linear basis functions for \(n^{+},n^{-}\), the DDR model results in 65 k plus 30 k internal degrees of freedom (DoFs), while the EQS with constant conductivity has 50k DoFs, reflecting the increased computational burden. Moreover, the DDR model is nonlinear. Therefore, it requires, e.g., the Newton-Raphson algorithm for the solution. Conversely, the EQS with constant \(\sigma\), described by Equation (5), is linear with respect to \(\varphi\). The steady-state trend of \(\varrho_{S}\) evaluated with the two models is reported in Figure 6. It can be seen that the two strategies are in good agreement. Under a high ionization rate, \(S=3\cdot 10^{9}\) m\({}^{-3}\) s\({}^{-1}\), the steady-state trend of \(\varrho_{S}\) is driven by the gas conductivity. As can be seen in Figure 7, the linear relation expressed by Equation (18), with \(\sigma_{G}=4.7\cdot 10^{-7}\) S/m, is not able to replicate the trend obtained with the DDR model due to the limitations of this model for high \(S\). As illustrated in Figure 8, if compared to the low-\(S\) regime, the \(n^{-}\) distribution drastically changes for a high \(S\), reflecting the spread of the population in the \(J\)-\(E\) plane. As can be seen in Figure 8, the density of negative ions in a high-ionization regime drastically differs from that in a low-ionization regime. Figure 5: Trend of \(\varrho_{S}(t)\) along the gas–solid insulator interface. The arc length is measured from the bottom to the top of the interface. Figure 6: Surface charge density \(\varrho_{S}\) at \(t=8000\) h along the gas–solid insulator interface obtained with the DDR and EQS with constant \(\sigma_{G}\). Low-ionization regime. The arc length is measured from the bottom to the top of the interface. The charge density in the solid insulator after \(t=8000\) h is compared for the low- and high-ionization rates. From the results shown in Figure 9, it can be seen that for the low-ionization case, the homo-charge is predominant, while hetero-charge dominates in the high-ionization regime. Figure 8: Negative ions’ distribution after \(t=8000\) h. (**a**) Low-ionization regime. (**b**) High-ionization regime. Figure 7: Surface charge density \(\varrho_{S}\) at \(t=8000\) h along the gas–solid insulator interface obtained with the DDR and EQS with constant \(\sigma_{G}\). High-ionization regime.