3 from MD simulations [14]. At this point we remark that in all our simulations we have found that the net charge induced by the external field is always located within 10 Å from the membrane surface and that the voltage drop across the membrane is very close in value to the external applied voltage, whatever the strength and frequency of the latter. This last observation is consistent with previ- ous studies [1]. Next, we fit these MD simulation results employing the equivalent circuit model given by Eq. (4). Using microscopic parameters, Eq. (4) can be rewritten as where A is the area of the membrane surface, µ is the ion mobility which is similar for both types of ions and we take to be 7 . 12 × 10 − 8 m $^{2}$V − $^{1}$s − 1 [15], e the ion charge, n the density of ions in the bulk, and τ = 2 R$_{s}$C$_{ss}$ is the relaxation time. The factor of 2 takes into account the conductivity of both channels (K + and Cl $^{−}$). Using dc- field simulations, we were able to extract τ . We do this by first applying a constant electric field (with the same magnitude as the amplitude of our ac-field simulations) which forces ions to accumulate in the vicinity of the membrane as it is shown in Fig. 1. The total net charge within 10 Å from the nanopore surface is integrated to obtain the capacitor charge [16]. Then, we turn off the field and monitor the decay of this charge back into the bulk (see Fig. 3). From this we obtain the relaxation time τ . We find that this time is slightly dependent on the amplitude of the initially applied electric field, be- coming slightly longer as the applied field decreases. For voltages as low as 5 V across the membrane we get from our simulations a relaxation time τ ∼ 75 ps [17]. Solving Eq. (5) for Q ( t ) with an ac electric field E ( t ) = E$_{0}$ sin ( ωt ) of amplitude E$_{0}$ and angular frequency ω = 2 πf , we obtain the long-time limit solution: To see this, we write the external voltage across the whole system as V ( t ) = V$_{0}$ sin ( ωt ) , with V$_{0}$ = − E$_{0}$d where d is the size of the simulation cell along the direc- tion of the electric field. From C = Q ( t ) /V ( t ) and (6) we then obtain the long-time limit of the capacitance: This represents the main result of our paper. It shows that the capacitance of the whole nanopore system is history-dependent, diverges when the voltage crosses its zero value, and acquires negative values within a certain voltage range. This is plotted in Fig. 4 where both the charge on the capacitor and the capacitance are plot- ted as a function of voltage across the capacitor. From Fig. 4 it is evident that at low frequencies, this ionic memcapacitor behaves almost like a regular capacitor. At higher frequencies, memory effects in the capacitance manifest in a hysteresis loop characteristic of memcapac- itors. However, unlike typical memcapacitors that show a pinched hysteretic loop [4], ionic memcapacitors have a non-vanishing (diverging) zero-bias capacitance. Memory at small frequencies — At small enough fre- quencies, there will be enough ion transport through the pore to play a role in the dynamics. In fact, it is no longer correct to assume that the charge associated with C$_{e}$$_{−}$$_{s}$ is the same as the charge associated with C$_{ss}$ . Us- and δ = tan − $^{1}$( ωτ ) . Using the appropriate values of A , n , and τ as obtained from the numerical simulations, this model agrees very well with the results of our MD simulations as it is evident from Fig. 2. In particular, it is clear that the net charge and voltage on the capacitor are phase shifted by the amount δ . This means that when the voltage is zero the charge on the capacitor is not necessarily zero, and vice versa. This is shown in which amounts to a sine function with a phase shift, namely Q ( t ) = Q$_{0}$ sin ( ωt − δ ) , where Q ( t ) t →∞ = 2 AµenE$_{0}$ sin ( ωt ) $^{(}$1 + 1 ω $^{2}$τ2 $^{)}$ω $^{2}$τ − 2 AµenE$_{0}$ cos ( ωt ) $^{(}$1 + 1 ω $^{2}$τ2 $^{)}$ω (6) Q$_{0}$ =   ( 2 AµenE$_{0}$ $^{(}$1 + 1 ω $^{2}$τ2 $^{)}$ω$^{)}$$^{2}$ +$^{(}$ 2 AµenE$_{0}$ $^{(}$1 + 1 ω $^{2}$τ2 $^{)}$ω $^{2}$τ ) 2   1 / 2 (7) C ( t ) t →∞ = 2 Aµen $^{(}$1 + 1 ω $^{2}$τ2 $^{)}$ω $^{2}$τd − 2 Aµen $^{(}$1 + 1 ω $^{2}$τ2 $^{)}$ωd cot ( ωt ) , (8) d Q ( t ) d t = 2 eAµnE ( t ) − Q ( t ) τ , (5) Figure 3. (Color online) Net charge of the positively charged side of the the capacitor vs. time when a constant 20 V ap- plied voltage responsible for the accumulation of the charges on the surface of the pore is turned off. A decay timescale of 75 ps is obtained by fitting the simulation results with an exponential decay curve, i.e., Q ( t ) = Q$_{max}$e − $^{t/τ}$. Due to the nearly equal mobilities of K + and Cl − [15], we find similar results for the negatively charged side and different field strengths as well. This is shown in the inset with initial voltage of 5 V. Fig. 4(a) where the net charge is plotted as a function of the voltage across the capacitor. This gives rise to diverging and negative values of capacitance.