## 3 Results and Discussion
### Differential Capacitance Calculation and Analysis  
Apparently, the abovementioned different characteristics of ion distributions near the non-rough and rough electrodes could affect the capacitance properties of the EDL, which can be represented by the C-V curve (i.e., differential capacitance versus the potential drop across the EDL) of the EDL. To obtain the C-V curve, it is necessary to firstly find the total potential (\(\Phi\) = \(\Phi_{\sigma}\) + \(\Phi_{IL}\)) distribution of the whole simulation system along the \(z\)-direction. Herein, \(\Phi_{\sigma}\) = \(\sigma\)z/\(\varepsilon\) is produced by the charge on the electrode, where \(\varepsilon\) is the effective dielectric constant of the electrolyte. \(\Phi_{IL}\) is produced by the ionic liquid, which can be gained by integrating the Poisson equation [39]:  
\[\frac{d^{2}\Phi_{IL}(z)}{dz^{2}}=-\frac{\rho_{e}(u)}{\varepsilon} \tag{1}\]  
where \(\rho_{e}(u)\) = \(e(\rho_{cat}-\rho_{ani})\) is the space charge density along the surface normal direction (\(\rho_{cat}\) and \(\rho_{ani}\) are the ion concentrations of [EMIM]\({}^{+}\) cautions and [BF\({}_{4}\)]\({}^{-}\) anions, respectively, and e is the elementary charge), and \(\varepsilon\) = \(\varepsilon_{0}\varepsilon_{r}\) is the effective dielectric constant of the electrolyte (\(\varepsilon_{0}\) and \(\varepsilon_{r}\) are the permittivity of vacuum and relative dielectric constant, respectively). Considering that the dielectric constant in the compact layer can be nearly ten times smaller than that in the bulk phase [25; 40], we chose \(\varepsilon_{r}\) = 7.8  
Figure 7: Total interaction energies of the [EMIM]\({}^{+}\) ion and [BF\({}_{4}\)]\({}^{-}\) ion at different positions on the neutral electrode surface. for the [EMIM]\({}^{+}\)/[BF\({}_{4}\)]\({}^{-}\) electrolyte. With the boundary condition \(\Phi(0)=0\) and \(d\Phi/dz=0\) in the bulk phase of electrolyte, the total potential distribution along the \(z\)-direction can be expressed as:  
\[\Phi(z)=-\frac{1}{\varepsilon}\int_{0}^{z}{(z-u)\rho_{e}(u)du+\frac{\sigma}{ \varepsilon}z} \tag{2}\]  
When the potential distribution is obtained according to Equation (2), the potential drop across the EDL (\(U_{EDL}\)) can be calculated as \(U_{EDL}=\Phi_{e}-\Phi_{b}\), where \(\Phi_{e}\) is the potential on the electrode and \(\Phi_{b}\) is the potential in the middle of the electrolyte phase. In this way, \(U_{EDL}\) values on the non-rough electrode and the rough one are calculated as shown in Figure 8. Based on the data points of \(\sigma\) versus \(U_{EDL}\), the \(\sigma\)-\(U_{EDL}\) curve is obtained by fitting these points with a fifth-order polynomial (see Figure 8). As shown, the potential drop across the EDL (\(U_{EDL}\)) is not zero but a positive value at \(\sigma=0\) e/atom. As discussed above, this phenomenon must be generated by the first layer of ions in the EDL on the neutral electrode surface. Accordingly, the C-V curves can be obtained by analytically differentiating the polynomial according to the equation \(DC=d\sigma/dU_{EDL}\), and are depicted in Figure 9 for the non-rough and rough electrodes. As shown, the C-V curves on the non-rough and rough electrodes both exhibit the double-hump shape. That is, a minimum point appears on the C-V curves near the potential of zero charge (PZC), while two maximum ones appear at lower and higher values of \(U_{EDL}\). As discussed above, it is the accumulation of ions near the negatively and positively charged electrode surface that results in the maximum capacitance at negative \(U_{EDL}\) and the one at positive \(U_{EDL}\). However, with the increasing concentration of ions near the charged electrode surface, the accumulation of ions at the charged electrode surface tends to be weakened when the electrode is further charged. As a result, the capacitance climbs over two maximum points and decreases in turn, forming the double-hump-shaped C-V curves.