# Ion Current Rectification in Extra-Long Nanofunnels
## 3 Results and Discussion
### Accumulation
Figure 4: K\({}^{+}\) (red line) and Cl\({}^{-}\) (dashed blue line) ion concentration along the longitudinal section of the nanochannel as a function of the applied voltage between the reservoirs (C\({}_{\text{Bulk}}\) = 1 M = 10\({}^{3}\) mol/m\({}^{3}\)). Figure 5: 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\({}_{\text{Bulk}}\) = 10\({}^{-5}\) M = 10\({}^{-2}\) mol/m\({}^{3}\)). The independence of the concentration distribution from the sign of the charge is progressively broken by varying the applied voltage from \(-\)10 (Figure 6a) up to \(+\)10 V (Figure 6d). At \(-\)10 V, both C(K\({}^{+}\)) and C(Cl\({}^{-}\)) are up to 40 times larger than C\({}_{\rm Bulk}\) in the left half of the nanochannel, while both concentrations decrease to a very low value in the right half, towards the nanochannel end. This is due to the ion depletion area, which forms in the anodic reservoir around the nanochannel end. At low C\({}_{\rm Bulk}\) and high electric potential difference between the reservoirs, the depletion area propagates into the channel, forcing the ions to accumulate in the left half of the nanochannel. By decreasing the absolute value of the applied voltage, the ion distribution becomes charge sign-dependent; while K\({}^{+}\) ions are uniformly distributed in the channel and reach a plateau for \"zero\" and positive voltages, Cl\({}^{-}\) are progressively dropped out (Figure 6d-f). The dependency on the ion charge sign at a positive applied voltage (\(+\)1 V) is clearly visible in the colored plot of Figure 7: C(K\({}^{+}\)) increases in the nanochannel, while for Cl\({}^{-}\) ions, the enrichment area in the cathodic side does not penetrate into the nanochannel. As we will also see in the following, in the nanochannel C(K\({}^{+}\)) is constant along longitudinal sections, whereas it increases towards the negatively charged surfaces. The asymmetric response with respect to the applied voltage suggests the use of the device as a rectifier at low C\({}_{\rm Bulk}\), with an \"forward state\" for negative voltages and a \"reverse state\" for positive ones. Later on, we will further discuss this point by calculating the ionic current through the device as a function of V and C\({}_{\rm Bulk}\). These results are mainly due to the large value of the Debye length for low concentration. In fact, for C\({}_{\rm Bulk}\) = 1 M, \(\lambda_{\rm D}\) = 0.3 nm, this is negligible compared with the nanochannel half-width (40 nm), whereas at C\({}_{\rm Bulk}\) = 10\({}^{-5}\) M, \(\lambda_{\rm D}\) = 96 nm, this is even larger than the width of the nanochannel itself. The increase in the Debye length when the ionic strength of the solution decreases determines an overlapping of the two EDLs in the nanochannel for a concentration lower than 10\({}^{-3}\) M.  
Figure 8 clearly shows that, at half the length of the nanochannel for C\({}_{\rm Bulk}\) = 10\({}^{-5}\) M, the K\({}^{+}\) and Cl\({}^{-}\) concentrations along with the transversal section change continuously from the center to the walls of the channel, with an increase and decrease by a factor 2 for C(K\({}^{+}\)) and C(Cl\({}^{-}\)), respectively. Moreover, at such a low concentration, the profile depends strongly on the applied voltage. The dependence of the ion accumulation/depletion on C\({}_{\rm Bulk}\) is well described by the graphs in Figure 9, in which the percentage increase in the ion concentration, C\({}_{\rm i}\)/C\({}_{\rm Bulk}\), is plotted along the funnel for different C\({}_{\rm Bulk}\) and applied voltages V. The accumulation effect for negative voltage (\(-\)1 V) is strongly enhanced at low C\({}_{\rm Bulk}\) (magenta and blue curves), but almost independent of the ion charge sign at any C\({}_{\rm Bulk}\) (Figure 9a, b). Only at very low C\({}_{\rm Bulk}\) (10\({}^{-5}\) M) is C(K\({}^{+}\)) larger than C(Cl\({}^{-}\)) at the funnel end.