Figure 4. (Color online) Upper panel: net charge Q versus a periodic voltage of amplitude V$_{0}$ = 1 V and different fre- quencies as obtained from Eq. 6. The inset shows the same quantity for 2 GHz and V$_{0}$ = 20 V compared with the charge obtained directly from our MD simulations. Lower panel: ca- pacitance versus a periodic voltage of frequency f = 10 MHz and amplitude V$_{0}$ = 1 V as obtained from Eq. 8. In the in- set we show the same quantity for f = 2 GHz and V$_{0}$ = 20 V compared with the capacitance obtained directly from our MD simulations. It is seen that C can be both negative and diverges as the voltage approaches zero. The arrows indicate the direction the voltage is swept in time and the numbers show the order in which the trace is generated. ing Kirchoff’s loop rule, we get an equation similar to Eq. (3): except now we differentiate between the two charges, Q$_{e}$$_{−}$$_{s}$ (on the external plates) and Q$_{ss}$ (on the pore). Applying Kirchoff’s junction rule we then get where due to C$_{ss}$ and R$_{ss}$ being in parallel, V$_{ss}$ is sim- ply the voltage on the C$_{ss}$ capacitor, Q$_{ss}$/C$_{ss}$ , and we can approximate the conductance across the pore as 1 /R$_{ss}$ = 2 A$_{p}$enµ/L$_{p}$ where A$_{p}$ = πr 2 $_{p}$, with r$_{p}$ the pore Experimental test — The results presented in this pa- per show that there are two very distinct memory regimes for nanopores in solution. To observe the first memory mechanism due to the polarization of the ionic solution, ideally, one would need high frequencies (on the order of 1GHz). On the other hand, much smaller frequencies (on the order of 1KHz) are necessary to observe the addi- tional memory effect due to transport through the pore. This indicates that the second memory mechanism would be the easiest one to detect. However, in both cases we have shown that the capacitance acquires diverging (and For an ac field, this equation can be solved analytically but the solution is a bit involved. We then just plot it for different frequencies in Fig. 5. As anticipated, we see that direct transport across the pore leads to an additional memory mechanism at specific frequencies, much smaller than the ones due to the polarization of the ionic solution. radius, e the electronic charge, n the ion density, µ the ion mobility, and L$_{p}$ the length of the nanopore. We ap- proximate C$_{ss}$ as that of a parallel plate capacitor, i.e., C$_{ss}$ = ϵA/L$_{p}$ with ϵ the permittivity of Si$_{3}$N$_{4}$, and A is again the area of the membrane surface. We can solve Eq. 9 for Q$_{ss}$ and plug it into Eq. 10 to arrive at the following second order equation for Q$_{e}$$_{−}$$_{s}$ Figure 5. (Color Online) Memcapacitive effects due to ionic transport across the pore. Main panel shows the charge on the external capacitor plates plotted versus the voltage across it for various frequencies of the electric field with V$_{0}$ = 1 . 0 V , A = 10 µm $^{2}$, L$_{p}$ = 25 Å, r$_{p}$ = 7 Å, ϵ = 7 . 5 ϵ$_{0}$ , C$_{ss}$ = ϵA/L$_{p}$ , and C$_{e}$$_{−}$$_{s}$ = 10 C$_{ss}$ for a 1 M concentration of ions. The inset shows the capacitance plotted versus the voltage for the same frequencies. Memory effects due to ionic transport across the pore occur at much lower frequencies than those due to the polarization of the ionic solution. 4 2 Q$_{e}$$_{−}$$_{s}$ C$_{e}$$_{−}$$_{s}$ + 2 R$_{s}$ dQ$_{e}$$_{−}$$_{s}$ dt + Q$_{ss}$ C$_{ss}$ = V ( t ) . (9) dQ$_{e}$$_{−}$$_{s}$ dt = dQ$_{ss}$ dt + V$_{ss}$ R$_{ss}$ . (10) 2 R$_{s}$C$_{ss}$ d $^{2}$Q$_{e}$$_{−}$$_{s}$ dt 2 +$^{(}$1 + 2 C$_{ss}$ C$_{e}$$_{−}$$_{s}$ + 2 R$_{s}$ R$_{ss}$$^{)}$ dQ$_{e}$$_{−}$$_{s}$ dt +2 Q$_{e}$$_{−}$$_{s}$ R$_{ss}$C$_{e}$$_{−}$$_{s}$ = V ( t ) R$_{ss}$ + C$_{ss}$ dV ( t ) dt (11)