# Ion Current Rectification in Extra-Long Nanofunnels
## 3 Results and Discussion  
### Accumulation  
By varying the boundary conditions for the Poisson equation at wall 1 of the _cis_-reservoir (wall 1 of the _trans_-reservoir is grounded, i.e., V = 0 V), we calculated C(K\({}^{+}\)) and C(Cl\({}^{-}\)), the concentration of K\({}^{+}\) and Cl\({}^{-}\) ions, respectively, inside and at the interface of the structure as a function of the voltage applied. Positive currents (\(I>0\)) flow from the wide opening towards the narrow opening of the nanochannel. With 1 M concentration in the two reservoirs (C\({}_{\rm Bulk}\)), an ion accumulation occurs in the funnel for negative voltages, with a peak increase in the concentration of about 0.8% in the proximity of the nanochannel entrance (Figure 2). On the other hand, an ion depletion (same \"intensity\" of the accumulation) occurs for positive voltages. Interestingly, at C\({}_{\rm Bulk}\) = 1 M the accumulation (depletion) is independent of the sign of ion charge (Figure 2a,b). This phenomenon is due to the presence of surface charges on the walls of the funnel and nanochannel and, consequently, to the formation of an Electric Double Layer (EDL), whose thickness, called the Debye length, \(\lambda_{D}\), can be estimated as follows  
\[\lambda_{D}=\sqrt{\frac{\varepsilon k_{B}T}{2C_{bulk}eN_{A}}} \tag{1}\]  
where \(k_{B}\) is the Boltzmann constant, \(T\) the absolute temperature, \(C_{bulk}\) the ionic concentration of the solution, \(N_{A}\) the Avogadro number, \(\varepsilon\) the permittivity of free space, and \(e\) the elementary charge. Figure 1: Sketch of the system geometry used for the FEM calculations. The red numbers correspond to walls with the same boundary conditions. Indeed, simulations performed considering no surface charge, i.e., \(\sigma\) = 0 C/m\({}^{2}\), present a constant ion concentration along the entire device (not shown here). However, the accumulation effect at a high concentration (C\({}_{\rm Bulk}\) = 1 M) is almost negligible, since the EDL is about 0.3 nm, as evidenced by the constant concentration along the transversal section in the middle of the nanochannel for both ions at any applied voltage (Figure 3). Only very close to the walls (distance less than 2 nm) does an increase in the K\({}^{+}\) ion concentration (a decrease for Cl\({}^{-}\)) occur. Figure 3: K\({}^{+}\) (**a**) and Cl\({}^{-}\) (**b**) ion concentration along the transversal section at the middle of the nanochannel as a function of the applied voltage between the reservoirs (C\({}_{\rm Bulk}\) = 1 M = 10\({}^{3}\) mol/m\({}^{3}\)). Figure 2: K\({}^{+}\) (**a**) and Cl\({}^{-}\) (**b**) ion concentration along the longitudinal section of the funnel and nanochannel as a function of the applied voltage between the reservoirs (C\({}_{\rm Bulk}\) = 1 M = 10\({}^{3}\) mol/m\({}^{3}\)). Figure 4 shows C(K\({}^{+}\)) e C(Cl\({}^{-}\)) along the longitudinal section of the nanochannel. The graphs clearly indicate that there is no difference in the concentration of the two species at any applied voltage, since all curves are superimposed. The accumulation effect is strongly dependent on C\({}_{\text{Bulk}}\). By decreasing by 5 orders of magnitude the ion concentration (C\({}_{\text{Bulk}}\) = 10\({}^{-5}\) M), the behavior drastically changes (Figure 5): (i) the accumulation effect in the funnel is much stronger with the maximum concentration, which is 50 times larger than C\({}_{\text{Bulk}}\); (ii) the independency from the ion charge sign is broken for positive applied voltages. At C\({}_{\text{Bulk}}\) = 10\({}^{-5}\) M, both ions are accumulated in the funnel for negative voltages, while for \"zero\" and positive voltages, depletion of both ions occurs. Accumulation of K\({}^{+}\) ions for \"zero\" and positive voltages occurs only in the nanochannel, whereas C(Cl\({}^{-}\)) is very low. A zoom-in of the ion concentration as a function of the position along the longitudinal section of the nanochannel shows this clearly (Figure 6).