# Ion Current Rectification in Extra-Long Nanofunnels
## 3 Results and Discussion
### Rectification  
As previously mentioned, at low \(C_{\text{Bulk}}\) the asymmetric response of the device for positive and negative applied voltages suggests the possibility to use it as a current rectifier. In order to estimate this effect, we consider the current density \(J\) at the point (\(x\),\(y\)) due to the contributions of both ion species to be  
\[J(x,y)=-D_{K}\frac{dC_{K}}{dx}+D_{Cl}\frac{dC_{Cl}}{dx}-\frac{FD_{K}C_{K}}{RT} \frac{dV}{dx}-\frac{FD_{Cl}C_{Cl}}{RT}\frac{dV}{dx} \tag{2}\]  
where \(D_{i}\) is the diffusion coefficient and \(C_{i}\) is the ionic concentration of each ion species, \(F\) is the Faraday constant, and \(V\) is the electrical potential. The current which flows through section x of the device is thus:  
\[I(x)=F\int_{-y0}^{y0}J(x,y)dy \tag{3}\]  
where the integral is calculated along the channel width, i.e., from \(-y_{0}\) to \(y_{0}\). Since we performed 2D simulations, the current is calculated as the amount of charge per second flowing through a segment instead of a surface and is measured in A/m. For a discrete set of data, Equation (3) can be rewritten as  
\[I(x)=F\Delta y\sum_{y}J(x,y) \tag{4}\]  
where \(\Delta y\) is the sampling interval for \(C_{i}\), \(dC_{i}/dx\) and \(dV/dx\) along the y-direction. Figure 11 shows the plot of the current as a function of the applied voltage for different molarity of the KCl solutions. The currents have been normalized considering the concentration value in the reservoirs. At high \(C_{\text{Bulk}}\) (1 and 10\({}^{-1}\) M, black and dark yellow squares), the plot is symmetric with respect to the origin (i.e., no rectification). At \(C_{\text{Bulk}}=10^{-2}\) M (red squares), the rectification effect starts to be visible, whereas at 10\({}^{-3}\) M (green squares), the device clearly exhibits the characteristic behavior of a diode and the current flows through the device only for negative voltage. The rectification also occurs at 10\({}^{-4}\) M (blue squares), although at high voltage (\(-10\) V) the effect is less pronounced than expected,  
Figure 10: K\({}^{+}\) (**a**) and Cl\({}^{-}\) (**b**) concentrations in the nanofluidic device at C\({}_{\text{Bulk}}\) 10\({}^{-3}\) M (1 mol/m\({}^{3}\)), with -10 V applied between the reservoirs. Ion accumulation in the funnel for both ionic charges is clearly visible. while it decreases at \(10^{-5}\) M (magenta squares). The general trend of the ion current rectification ratio (\(\text{ICR}=\big{|}\text{I}(\text{V}_{\text{Bias}}<0)\big{|}/I(\text{V}_{\text{ Bias}}>0)\)) is shown in Figure 12. It is clear that the ICR increases by decreasing the solution concentration (from 1 M to \(10^{-4}\) M). Below a certain concentration, the ICR also depends on the applied voltage, while at \(\text{C}_{\text{Bulk}}=1\) M, the ICR is about 1 at any \(\text{V}_{\text{Bias}}\), at \(\text{C}_{\text{Bulk}}=10^{-3}\) M, the ICR is ca. 2.4 at 1 V, 11 at 5 V, and 20 at 10 V. Due to the limiting-current effect, at \(10^{-5}\) M the ICR drops to a value of ca. 5, independently from the applied voltage. Maxima of the rectification ratio as a function of the concentration were reported in both experimental and theoretical works [38; 39; 44; 55; 56; 57; 58]. A similar behavior was observed by our group in devices with similar geometry [27]. According to [PERSON] et al. [57], maxima occur at the bulk concentration at which the pore conductivity starts to differ significantly from the bulk, because of the contribution of surface-associated counterions to the current through the pore. For this bulk concentration, the conductivity of the pore in the \"forward state\" is significantly higher than in the \"reverse state\", resulting in a strong ICR. For lower concentrations, the conductivity values of the pore in the \"forward\" and \"reverse\" states are both dominated by the current arising from the surface charge, thus the ICR is weakened. Interestingly, [PERSON] et al. [43] and [PERSON] et al. [35] calculated ion current rectification for conical structures depending on length (up to 12 \(\upmu\)m).