Abstract
Electrolytefilled subnanometre pores exhibit exciting physics and play an increasingly important role in science and technology. In supercapacitors, for instance, ultranarrow pores provide excellent capacitive characteristics. However, ions experience difficulties in entering and leaving such pores, which slows down charging and discharging processes. In an earlier work we showed for a simple model that a slow voltage sweep charges ultranarrow pores quicker than an abrupt voltage step. A slowly applied voltage avoids ionic clogging and coion trapping—a problem known to occur when the applied potential is varied too quickly—causing sluggish dynamics. Herein, we verify this finding experimentally. Guided by theoretical considerations, we also develop a nonlinear voltage sweep and demonstrate, with molecular dynamics simulations, that it can charge a nanopore even faster than the corresponding optimized linear sweep. For discharging we find, with simulations and in experiments, that if we reverse the applied potential and then sweep it to zero, the pores lose their charge much quicker than they do for a shortcircuited discharge over their internal resistance. Our findings open up opportunities to greatly accelerate charging and discharging of subnanometre pores without compromising the capacitive characteristics, improving their importance for energy storage, capacitive deionization, and electrochemical heat harvesting.
Introduction
Electrolyteimmersed porous electrodes are an essential building block for many stateoftheart technologies in energy storage^{1,2,3,4}, energy harvesting^{5,6,7} and capacitive deionization^{8,9,10}. Of particular importance are electrical doublelayer capacitors (EDLCs), often called supercapacitors or ultracapacitors^{1,2,3,4,6,11,12}, which store energy by electrosorption of ionic charge into porous electrodes. Their performance is characterized by the energy density \({\mathcal{E}}\) and power density \({\mathcal{P}}\), which describe the amount of energy and the speed with which it can be supplied to an external load or device. Owing to their particular \(({\mathcal{E}},{\mathcal{P}})\) properties, supercapacitors find their way in applications that need higher power than delivered by batteries and more energy than stored in traditional dielectric capacitors. So far, the highest achievable capacitance^{13,14,15} and energy densities^{16} have been obtained with nanometresized pores. Hence, an extensive research effort has been dedicated to understanding the properties of such nanoporous supercapacitors^{12,17,18,19,20,21,22,23,24,25,26,27}, with an overarching goal being to maximize the stored energy density^{28,29,30,31,32,33,34,35} and the speed of charging and discharging^{36,37,38,39,40,41,42,43,44}.
It is often assumed that charging and discharging times are equal and proportional to the time constant \({\mathcal{E}}/{\mathcal{P}}\)^{2,45}. This reasoning probably originates from simple electronic RC circuits, which indeed display a charge/discharge symmetry (Supplementary Note S1). However, even for a simple EDLC model with planar electrodes, such charge/discharge symmetry is not present for applied potentials above the thermal voltage (≈25 mV at room temperature)^{46,47,48}. Molecular dynamics (MD) simulations have demonstrated that also nanopores charge and discharge in a dissimilar manner^{41}. The knowledge of discharging behaviour is thus insufficient for predicting charging behaviour and vice versa. Therefore, in this article, which centres around chargedischarge time optimization, we consider charging and discharging separately. Furthermore, as charge and discharge times are affected by many parameters, any optimization study should specify which parameters are varied and which are kept fixed. One could ask, for instance, given a charging or discharging procedure (voltage step or sweep, etc.), which supercapacitor charges/discharges the fastest? This is a problem of minimizing internal resistances via different pore shapes, lengths, etc.^{41,49}. An alternative question that one could ask is: given a supercapacitor, which timedependent cell voltage minimizes the time spent to charge it to a certain charge Q and to discharge the same supercapacitor to Q = 0? In this study, we focus on the second question, which has received much less attention in electrochemistry^{42}. In other contexts, optimization of timedependent protocols driving a system from an initial to a final state has been investigated, for instance, to find shortcuts to adiabaticity in quantum systems^{50,51}, to engineer swift equilibration of Brownian particles^{52} and AFM tips^{53}, etc. Finding the voltage sweeps providing the shortest charging times is of obvious practical importance to supercapacitors. Finding optimal discharging sweeps, which we also set out to do, is relevant to electrochemical lowgrade heat harvesting^{7} and capacitive deionization^{8,9,10}, where operation speed may be more important than energy efficiency.
Recent MD simulations have shown that, when a potential difference is applied abruptly to a nanopore, counterions rush to the pore’s entrance and clog the pore, causing coion trapping deep in its interior and leading to an overall sluggish charging dynamics^{41,54,55}. Pore clogging can be avoided by applying the potential difference with a linear sweep, i.e., by varying it with a constant rate^{41} (Fig. 1e). We show in this study that charging can be made even faster than the fastest (optimized) linear sweep developed in ref. ^{41}, if the variation of the applied potential is matched to the actual rate of coion desorption. We propose a general expression for a nonlinear sweep function and consider two closedform approximations. We assess the benefits of such a nonlinear voltage sweep in MD simulations of a model supercapacitor and in experiments with novolacderived carbon electrodes featuring a narrow poresize distribution and mainly slitshaped pores (Fig. 1b and Supplementary Fig. S3).
Finally, for discharging, ref. ^{41} found that a step voltage unloads a supercapacitor faster than any finiterate linear potential sweep. Here we demonstrate, with MD simulations and experiments, that a supercapacitor can discharge even faster if we apply a nonlinear sweep consisting of a voltage inversion followed by a linear sweep to zero.
Results
Nonlinear voltage sweep to accelerate charging
To avert the catastrophic clogging of a nanopore subject to a large instantaneous voltage step, the timedependent potential difference should be increased gently, so that the coions can leave the pore before counterions clog it. For a tiny increment ΔU of the applied potential—much too small to yield pore clogging—the coion desorption time can be estimated as Δt = l^{2}/D_{c}, where D_{c} is the inpore coion diffusion constant and l the distance of the coion to the pore exit. Then the rate of an optimal linear sweep can be approximated by \({k}_{{\rm{opt}}}=\Delta U/\Delta {t}_{\max }\), with \(\Delta {t}_{\max }\) being the longest of these times^{41}. However, as Δt may differ from \(\Delta {t}_{\max }\) at different stages of charging, it may be possible to charge a nanopore faster than with the optimized linear sweep, by varying Δt, and hence the sweeping rate k, in the course of charging.
Derivation of the nonlinear sweep function
Before a potential difference is applied to a nanopore, the distribution of coions in a pore is homogeneous. We can split these coions into K_{0} (\(\approx {\rho }_{{\rm{c}}}^{0}w\ell a\)) imaginary ‘layers’ along the pore, where \({\rho }_{{\rm{c}}}^{0}\) is the inpore coion density at zero voltage, w is the pore width, ℓ is the pore length and a is the ion diameter, for simplicity assumed the same for cations and anions. We then apply the potential difference stepwise such that in equilibrium each voltage step would lead to exactly one fewer layer of coions in the nanopore. After each step, we keep the voltage constant until this one layer diffuses out. It is noteworthy that the system does not have to equilibrate fully during this time. For instance, at low voltages it might be sufficient that only the coions close to the pore exit redistribute in response to a voltage step. Provided the coion density is roughly homogeneous at the end of each waiting period, we can estimate the average distance over which the coions have to diffuse out after the nth voltage step as l = ℓ/(K_{0} − n + 1). The waiting time Δt_{n} after the nth step can now be estimated as the time needed by a coion layer to diffuse out of the pore:
where the coion diffusion coefficient D_{c} may depend on n. The total time t needed to charge a nanopore to a potential difference U is the sum of all time increments up to n = K_{0} − K(U), where K(U) is the equilibrium number of layers in the pore at U, i.e., \(t(U)=\mathop{\sum }\nolimits_{n = 0}^{n\,{<}\,{K}_{0}K(U)}\Delta {t}_{n}\). For example, taking a constant D_{c} =10^{−8} m^{2} s^{−1} and ℓ = 12 nm (K_{0} ≈ 8, cf. Fig. 2), we find t(U) ≈ 8 ns for the voltage U at which the pore becomes free of coions. In this sum, the waiting time Δt_{n} increased from Δt_{0} ≈ 0.18 ns after the first step to Δt_{7} ≈ 3.6 ns after the last step, a more than tenfold increase due to the increased distance l that the coions needed to cover to exit a pore with fewer and fewer coions.
Using the above voltage steps that each expels one layer of coions gives a crude procedure to charge a pore as fast as possible. We can refine this procedure by using many tiny voltage steps, each expelling some small amount of coions instead. Accordingly, we rewrite the sum over n as an integral and change the integration variable from n to voltage u, which yields:
Equation (2) is an implicit equation for the timedependent sweep function U(t) in terms of K(U). This equation provides the optimal sweep function for charging to an arbitrary potential difference below U. To demonstrate its applicability, we have numerically determined K(U) for the model supercapacitor shown in Fig. 1d with MD simulations; the results are shown by symbols in Fig. 2a. Assuming that D_{c} does not depend on the applied potential, we performed the integration in eq. (2) numerically using the composite trapezoidal rule (symbols in Fig. 2b; see also Figs. S5 and S6). The results demonstrate that U(t) can be varied quickly at early times, corresponding to low voltages, because there are still many coions close to the pore exit, and hence their ‘effusion’ path is relatively short (i.e., the path over which the coions need to diffuse to exit the pore). At intermediate times, corresponding to intermediate voltages, this effusion path becomes longer due to the reduced coion density and one must slow down the variation of the applied potential, to allow the coions to leave the pore. Finally, at long times (higher applied potentials), there are practically no coions left in the pore and the charging can proceed in an almost stepwise manner.
Stretched exponential approximation
To complement the above insights from the numerical evaluation of eq. (2), we approximated the simulation data of K(U) by a stretched exponential:
allowing us to perform the integral in eq. (2) analytically. Here, K_{0}, γ, and α are ℓdependent fitting parameters. We have used the dogleg leastsquare algorithm from scipy (https://scipy.org), to fit eq. (3) to the MD data. Figure 2a demonstrates that eq. (3) fits the MD data decently (solid line in Fig. 2a, see also Figs. S5, S6, and S7). Notably, we found that the fitting parameters α and γ depend weakly on the pore length, ℓ, whereas K_{0} varies with ℓ more significantly (viz. , K_{0} ~ ℓ, see Fig. S7). This is understandable as α and γ define the shape of K(U), which does not change appreciably with ℓ (Fig. S7), whereas K_{0} represents the number of coion layers in the pore at U = 0 V, which should indeed increase with the pore length.
Plugging eq. (3) into eq. (2) and assuming again that D_{c} is U independent, one finds:
where:
Inverting eq. (4) leads to:
Figure 2b shows that eq. (6) provides a good approximation to the nonlinear sweep function obtained by integrating eq. (2) numerically. For very long pores, \(\ell \gg {({\rho }_{{\rm{c}}}^{0}wa)}^{1}\), one has K_{0} ~ ℓ ≫ 1 and eq. (5) gives τ ~ ℓ^{2} in the leading order in ℓ. Then eq. (4) leads to charging times t(U) ~ ℓ at potential differences below threshold \({U}_{t}={\gamma }^{1}{(\mathrm{ln}\,{K}_{0})}^{1/\alpha }\) (at which the number of coion layers K(U) ≈ 1) and t(U) ~ ℓ^{2} at U ≫ U_{t}. The latter result is in line with the quadratic porelength scaling of the charging times of optimal linear sweep^{41}. For the porelengths considered in the simulations (ℓ = 12, 16, and 20 nm), the threshold voltage U_{t} varies slightly around 1.5 V and increases roughly as a square root of logarithm with increasing ℓ.
Linear approximation
Although the stretched exponential (eq. (3)) approximates the number of coion layers K(U) for all applied potentials considered (Fig. 2a), the resulting expression for the nonlinear sweep function, eq. (6), looks cumbersome. A useful approximation, likely most relevant to experimental systems^{24,26,56}, can be obtained for low voltages, U ≲ 1.3 V in Fig. 2a. We notice that K(U) varies roughly linearly in this regime, i.e.,
where γ and K_{0} are again ℓdependent fitting parameters. We have fitted eq. (7) to the MD data for U ≲ 1.3 V (dashed line in Fig. 2a and Fig. S8) using the dogleg leastsquare algorithm from scipy (https://scipy.org).
Plugging eq. (7) into eq. (2) and assuming D_{c} to be U independent, as before, one obtains:
where τ is given by eq. (5). Inverting eq. (8) yields a simple equation:
where U_{0} = (1 + K_{0})/(γK_{0}) and \(\tilde{\tau }=\tau /{K}_{0}\). It is interesting to note that the linear approximation for K(U), eq. (7), still yields a sweep function, eq. (9), which varies nonlinearly with time.
Figure 2b shows that eq. (9) approximates the nonlinear sweep function (obtained by numerical integration of eq. (2)) well at low voltages and poorly at high voltages. This is not surprising, as we fitted the linear approximation only in the lowvoltage regime.
For long pores, ℓ ≫ 1/(ρ_{c}wa), one has τ ~ ℓ^{2}, and eq. (8) gives t(U) ≈ [ℓ^{2}/(D_{c}K_{0})][γU/(1 − γU)]. Since K_{0} ~ ℓ and γ depends only weakly on ℓ (Fig. S8), the charging time increases proportionally to the pore length, ℓ. This linear ℓscaling is in line with the scaling found for the stretched exponential approximation at low applied potentials. Conversely, for the optimal linear sweep at high voltages the charging time scales as ℓ^{2}^{41}.
Results of MD simulations
To assess the benefits of the nonlinear sweeps, as compared to step voltage and optimal linear sweeps^{41}, we performed MD simulations, in which we applied these three different charging protocols to the model supercapacitor shown in Fig. 1d. In all simulations, we chose a large potential difference U = 3 V. The reason for this choice is that, at this U, coion trapping, which we aim to circumvent with slow voltage sweeps, occurs for relatively short, computationally feasible pores. For longer pores the trapping and pore clogging occur at lower potential differences (Fig. S4), suggesting that our approach applies to a wider voltage range.
For the nonlinear sweep function, we used eq. (6), as obtained by fitting the stretched exponential approximation to the MD data for K(U), the number of coion layers at equilibrium (Fig. 2). Equation (6) depends parametrically on τ, which, in turn, depends on the unknown coion diffusion coefficient D_{c} (eq. (5)). Accordingly, we treated τ as an optimization parameter and performed MD simulations of the outofequilibrium charging behaviour with the nonlinear sweep functions for several values of τ. Examples of U(t) for τ = 4 ns and τ = 12 ns are shown in Fig. 3a. Figure 3a also shows the accumulated charge and the number of counter and coions inside the pore as a function of time. Clearly, the nonlinear sweep provides faster charging than the step voltage for all τ considered. Only for some τ, the nonlinear sweep charges faster than the optimal linear sweep. When sweeps are too fast (small τ), coion trapping leads to sluggish desorption, as is the case for the stepvoltage charging or toofast linear sweeps (red and darkgreen curves in Fig. 3a). For sufficiently large τ, the coion desorption proceeds faster than for the optimal linear sweep, as anticipated (orange vs. lightgreen curves in Fig. 3a).
To determine the optimal τ, we studied the τ dependence of the coion desorption and counterion adsorption times, which are the times needed by the coion and counterion densities to reach their equilibrium values corresponding to the final applied voltage (these times are not the same^{41} and depend on charging protocol U(t); see Figs. S10 and S11). The desorption time increases drastically for small τ, which is due to coion trapping, and levels off for larger values of τ, as Fig. 3b demonstrates. The adsorption time increases roughly linearly with τ for τ > 8 ns, but increases abruptly for smaller values of τ (the dashed line in Fig. 3b). Figure 3b also shows that the optimal τvalue, τ_{opt}, i.e., the value that minimizes the charging time, can be found as a crossing point of adsorption and desorption times (there might be more than one crossing; clearly, τ_{opt} corresponds to the one providing the shortest time). For sufficiently slow charging, τ ≥ τ_{opt}, the charging times from the MD simulations are consistent with the charging times given by eq. (4), i.e., the pore charging completes at the same time as the voltage reaches its final value (τ = 12 ns in Fig. 3a and Fig. S11). For lower values of τ, the voltage varies too quickly, which leads to pore clogging and causes slow coion desorption (τ = 4 ns in Fig. 3a and Fig. S10).
Next, we performed MD simulations for a few pore lengths ℓ and determined τ_{opt} in each case. From the accumulated charge Q (obtained at τ_{opt}), we extracted the charging time, t_{charge}, which we defined as the time at which Q reaches 96% of the maximum charge capacity at a given applied potential (note that the so determined t_{charge} is only approximately equal to t(U), eq. (4), as we allowed for 4% tolerance in coion desorption, to be consistent with ref. ^{41}). The results are shown in Fig. 3c, along with the charging times obtained with the optimal linear sweep^{41}. The difference between the charging times for the linear and nonlinear sweep functions increases markedly with the pore length, ℓ. The data suggest that the nonlinear sweeps may provide significantly faster charging for longer pores.
Experimental results
To test our theoretical findings, we have performed charging experiments with a supercapacitor based on a symmetric twoelectrode setup and novolacderived activated carbons with wellcontrolled nanopores as the electrode material^{57} (Fig. 1a, b). As seen from the gas sorption data, the pore size distribution is very narrow with an average pore size of 0.68 nm and a total pore volume of 0.49 cm^{3}/g. We used EMIMBF_{4} ionic liquid as electrolyte (Fig. 1c). All experiments were conducted at room temperature (including discharging, see below). However, to check how our results depend on the IL conductivity, we carried out a series of additional experiments at an elevated temperature, at which the bulk conductivity was doubled (Fig. S12). At both temperatures we observed qualitatively the same results (Fig. 4 and Fig. S13).
We first applied a steplike cell voltage to the supercapacitor and measured how the accumulated charge varies over time (the solid red lines in Fig. 4). We found that the charge can be fitted decently by the sum of two exponents^{38}, \(Q(t)={Q}_{\infty }[1{a}_{1}\exp (t/{\tau }_{1}){a}_{2}\exp (t/{\tau }_{2})]\) (for the fitting parameters and the plot see Fig. S14). The slow timescale τ_{2} ≈ 45 min is in agreement with model predictions saying that supercapacitors charge at late times with the timescale τ = α(L/2 + H)^{2}/D ≈ 17 min^{58}, where L ≈ 150 μm is the electrode separation (which in the experiments is the thickness of the electrode–electrode separator), H ≈ 109 μm is the electrode thickness, D ≈ 10^{−11} m^{2}/s the bulk diffusion constant of EMIMBF_{4}. The salt concentration prefactor α = 0.59 was determined for the case of a model RTIL with ionic radii the same as in our simulations (0.5 nm) (private communication with Cheng Lian). As our supercapacitor model contains just two pores and a nanometresized reservoir (Fig. 1d), and hence does not account for the multiscale nature of real supercapacitors, this slow diffusive response is absent in our MD simulations. Conversely, the model of ref. ^{58} does not describe the dynamics of finite size ions in nanopores in as much detail as our simulations do, and hence, ref. ^{58} could not account for the pore clogging effects central to our article.
We next studied how this abrupt stepvoltage charging compares with slower voltage sweeps. We generated nonlinear sweep data in a computer for a few values of τ according to eq. (6), similarly as in the simulations. The nonlinear sweeps were produced by a piecewise approximation via linear functions (dashdot lines in the top plot of Fig. 4). To study the linearsweep charging, we used a ‘singlecycle’ voltammetry; the single run, rather than cycling, was necessary to ensure that the supercapacitor was fully discharged initially. The sweep rates k were chosen such that the times τ_{k} = U/k (U = 3 V is the applied cell voltage) were comparable to the values of τ used in the nonlinear sweeps (in the voltammetric experiments, the cell voltage was swept to zero and the device was allowed to discharge after reaching the final value of 3 V. For clarity, we show only the charging parts of these Q(t) curves in Fig. 4.)
Figure 4 shows that the voltage step is overtaken by all four slower applied potentials. This constitutes the first experimental verification of ref. ^{41}, which suggested that a stepvoltage does worse than linear sweep charging. We note, however, that both the linear and nonlinear sweeps have not been optimized and hence, unlike in the simulations (Fig. 3), the charge in Fig. 4 continues to grow after the cell voltage reaches its final value.
Although our experiments have demonstrated that a slow variation of applied potential can make charging faster than the stepvoltage, they were inconclusive regarding the comparison of the linear and nonlinear sweeps (we note, however, that the nonlinear sweep with τ = 20 min performed slightly better than the corresponding linear sweep with k = 2.5 mV/s, which is consistent with the simulations). This is likely because we have not reached the optimal sweep rate, which appeared to be slower than 0.1 mV/s (Fig. S15), implying that the time needed to fully charge the supercapacitor was >8 h. Thus, the voltage sweeps in the experiments operated only in short times (compare with τ = 4 ns in Fig. 3a). Nevertheless, both the simulations and the experiments clearly demonstrate that ‘slow’ voltage sweeps have the potential to speed up charging.
Accelerating discharging by voltage inversion
Turning to the discharging of supercapacitors, we are interested in the quickest discharging voltage sweep U(t) that takes a supercapacitor from a charged state at applied potential U_{ch} > 0 V to the fully discharged state at U = 0 V. It has been suggested^{41} that the optimal linear discharging sweep is as fast as possible, i.e., via a step to U = 0 V. This is different from optimal charging with a gentle voltage sweep. For step discharging the ions diffuse passively in and out of the pore, which is in contrast to active migration in the bulk of a supercapacitor as driven by the applied potential during charging. Hence, there is no danger of clogging the pore entrance either with coions or counterions. However, could discharging be even faster than what is achieved by a step voltage? This may be expected for an EDLC with planar electrodes, which has been successfully modelled for low voltages through an equivalent RC circuit^{59} (see also ref. ^{49}). The dielectric capacitor of such a circuit can be discharged ‘instantaneously’ by applying a negative deltalike spike (as obtained from the mathematical model, ignoring dissipation, see Supplementary Note S1), suggesting that an analogous spedup discharging could be achieved for the corresponding doublelayer capacitor (note, however, that the applicability of the circuit model to doublelayer capacitors is questionable for such a spike). For a nanoporous supercapacitor, it is unclear how a negative applied potential could speed up counterion desorption, as the electric field vanishes inside nanopores^{60}. But there might well be poreentrance effects. For instance, applying a negative potential difference to a nanopore may deplete counterions from the pore entrance area, leaving the counterion desorption akin to an expansion of a gas into a vacuum, which is faster than into a finite density gas. However, applying a toonegative potential difference may clog the pore entrance with coions, in the same way as the attracted counterions impede coion desorption in the charging process. To discern how these two competing phenomena play out, we have studied the behaviour of our supercapacitors upon discharging via a voltageinversion discharging sweep (Fig. 1e):
where k_{inv} is the slope with which the voltage approaches U = 0 V after applying a voltage inversion of magnitude U_{inv} < 0 V at time t = 0, and τ_{inv} = −U_{inv}/k_{inv} is the time at which U = 0 V.
Results of MD simulations
Figure 5 shows examples of U(t) given by eq. (10) for a few values of U_{inv} and k_{inv}, along with the corresponding numbers of co and counterions obtained by MD simulations. We first analyse how this discharging behaviour depends on k_{inv} at fixed U_{inv}. When k_{inv} is too small, i.e., when the applied potential reaches U(t) = 0 too slowly, the co and counterions greatly overshoot their final values. This early overshoot is followed by a slow relaxation to equilibrium, evidently giving long discharging times (Fig. 5a–c). When k_{inv} is too high, that is, when the application time of the (varying) negative potential difference is too short (small τ_{inv} = −U_{inv}/k_{inv}), then co and counterions ‘undershoot’ and discharging proceeds similarly as in the case of step voltage (Fig. 5g–i; step voltage not shown, but cf. Fig. 6a). This suggests that for each inversion voltage, U_{inv}, there must be a single optimal k_{inv} that minimizes the discharging time (for this U_{inv}).
We then optimized discharging over k_{inv} for a set of inversion voltages U_{inv}. The optimal k_{inv} was determined as k_{inv} providing the minimum discharging time, defined as the time at which the charge Q vanishes and exhibits only small fluctuations around Q = 0 within 5% of the initial charge (note that it is possible that the charge crosses zero and then decays to Q = 0 from the other side). Simulation results for fixed U_{inv} are shown in Fig. 6a, where we compare a few voltageinversion discharging curves with the stepvoltage discharging. Similar to the number of co and counterions, also the charge overshoots when the inversion voltage is applied for too long (k_{inv} = 2.5 V/ns in Fig. 6a, corresponding to τ_{inv} = 1 ns, see also Fig. 5a–c). Conversely, for high k_{inv}, discharging proceeds akin to the charge response to a stepvoltage, albeit slightly faster (compare k_{inv} = 10 V/ns and Step in Fig. 6a). Remarkably, the optimal k_{inv} (k_{inv} ≈ 5.5 V/ns in Fig. 6a) provides a fewfold shorter discharging time, as compared to the step voltage (~0.4 and 1.5 ns, respectively).
In Fig. 6b we plot discharging times, calculated at optimal k_{inv} values, as a function of the inversion voltage U_{inv}. This figure clearly demonstrates that there is a global minimum in the discharging times, obtained at U_{inv} ≈ −2.5 V for our model supercapacitor charged at U_{ch} = 3 V (it is noteworthy that an optimal pair (U_{inv}, k_{inv}) depends on U_{ch}). For higher U_{inv} and high k_{inv}, the inversion voltage induces strong overshooting (Fig. 5e, f), whereas for lower U_{inv} the system ‘undershoots’ and discharging proceeds similarly as in the case of step voltage (Fig. 5d, e).
We thus conclude that our voltage inversion scheme (eq. (10)), when properly optimized, provides fewfold lower discharging times than a voltage step. Although we have used samesize ions in our simulations, we expect similar behaviour for cation and anions with different sizes. In this case, an optimal (U_{inv}, k_{inv}) pair might be different for the cathode and the anode and one may need to compromise the speed of discharging at one of the electrodes. In practice, U_{inv} and k_{inv} must be optimized for the entire supercapacitor.
Experimental results
With the same supercapacitor based on the novolacderived porous carbons (Fig. 1a, b), we carried out experiments to validate the MD predictions that voltage inversion can accelerate discharging. In Fig. 7 we compare a stepvoltage discharge (U_{inv} = 0 V) with two voltage inversions. We chose U_{inv} = −2.5 V based on the MD simulation results (Fig. 6), but the values of k_{inv} for the experimental setup had to be taken much smaller than k_{inv} used for the single nanopore of the MD simulations. Figure 7 demonstrates that the voltage inversions discharge the supercapacitor much faster than the voltage step.
For the voltage inversions, the charge on the electrode becomes negative at intermediate times, after which it becomes positive again. We have not seen this qualitative feature so pronounced in the MD simulations (compare with the green curve in Fig. 6a). One likely has to account for ionic currents in the quasineutral macropores of the porous electrodes^{58}, which are ignored in our MD simulations.
Discussion
We have investigated how charging and discharging times of supercapacitors can be minimized by judiciously choosing timedepending sweeping rates of potential difference. Previous work has shown by MD simulations that when a stepvoltage is applied to an electrode with subnanometre pores, the quickly adsorbed counterions clog the pore entrance and lead to coion trapping, causing sluggish charging dynamics^{54,55}. Such clogging can be avoided by applying the potential difference slowly via an optimized linear sweep^{41}. Here we demonstrated that one may achieve even faster charging by adjusting the rate of voltage variation to the rate of coion desorption. We presented a general expression for such a nonlinear voltage sweep U(t) (via inversion of eq. (2)). Unlike linear sweeps, which require a separate optimization for each potential difference, the proposed U(t) provides the optimal charging path for all voltages below U. Computing U(t) requires the knowledge of the inpore counterion density, which is not straightforward to measure. We therefore considered two closedform approximations, eqs. (6) and (9). These approximate expressions depend on three and two parameters, respectively, which in practical applications can be treated as optimization variables. With MD simulations, we showed that the optimized nonlinear sweep can indeed provide a significantly faster charging (Fig. 3). The gain in the charging time over the linear optimal sweep increases for increasing pore length (Fig. 3c), which suggests that the nonlinear sweeps can be particularly relevant for realistically long nanopores. We showed experimentally with novolacderived porous carbons that slow voltage sweeps indeed provide faster charging (Fig. 4), but more work needs to be done to differentiate the linear and nonlinear protocols in the regime of full charging.
For discharging we have found that an uncharged state could be reached much faster than diffusively. This might seem surprising given that no electrostatic driving force acts on the inpore ions. However, applying a ‘negative’ potential difference to an electrode (i.e. , opposite in sign to the potential of a charged state) removes counterions from the poreentrance, effectively speeding up their desorption while also accelerating the adsorption of bulk coions into the pores (Fig. 5). We optimized the discharging time of such a twoparameter voltage inversion procedure, in which the applied potential steps to a negative value and returns to zero with a linear voltage increase (eq. (10)). Using MD simulations, we observed a fewfold decrease in discharging times, as compared to the purely diffusive stepvoltage discharging (Fig. 6). We experimentally confirmed these MD findings with the same novolac electrodes. Although we assessed this procedure with subnanometre pores and neat ionic liquids, it shall work for mesoporous electrodes as well as dilute electrolytes, because it is based on temporary speeding up of ion adsorption/desorption by applying a transient potential difference (Supplementary Note S1C). The finding that discharging can be accelerated may find a useful application in capacitive deionization, where for the production of potable water via ion electrosorption, the operation speed is a very important factor.
Methods
Simulations
We have carried out MD simulations with the simulation package ESPResSo (version 3.3.1)^{61,62,63,64}. The electrode geometry was built with carbon particles based on a triangular mesh of the slitshaped surface, resulting in a hexagonal carbon structure. Throughout the study, we used the following parameters: Pore entrance radii of 4 Å, pore closing radii of 2 Å, an accessible pore width of 0.6 nm, and an electrode separation (or bulk size) of 8 nm. We used pore lengths of 12, 16 and 20 nm for our study of the charging behaviour, whereas a single pore length of 12 nm was used in the discharging part.
We used the ICC* algorithm^{65} to carry out constantpotential simulations. The ICC* method recalculates the charges on the carbon atoms every time step (2 fs) to model metallic boundary conditions and charge induction caused by ions. This was supplemented by the precalculated electrostatic potential ϕ (varying in space) due to the voltage applied between the two electrodes; ϕ was obtained by solving the Laplace equation^{66} of the respective electrode geometry with a reference potential drop (between the electrodes) of 1 V, which was rescaled in accord with a timedependent potential difference during a simulation.
To gain computational efficiency and consistency with previous studies^{41,55,67,68,69,70,71}, we used the Weeks–Chandler–Anderson (WCA) potential for the interactions between all particle species. We chose the parameters of this softcore repulsive interaction as follows: σ_{c} = 3.37 Å and ϵ_{c} = 1 kJ/mol (carbon atoms) and σ = 5 Å and ϵ = 1 kJ/mol (ions), i.e., the monovalent ions were treated as symmetric WCA particles. The results of ref. ^{41} show that coion trapping and pore clogging occur for both sizesymmetric and sizeasymmetric ions and hence linear voltage sweeps can accelerate charging in both cases. Likewise, we expect that nonlinear sweeps can speed up charging for asymmetric ions but we leave such studies to further work.
We used the velocityVerlet algorithm^{72} in the NVT ensemble to propagate the system and a Langevin thermostat at temperature T = 400 K and damping constant ξ = 10 ps^{−1}. The thermalization dissipates the temperature increase due to Ohmic losses that appear during charging^{73,74}. This also alters the dynamics of the system, so the choice of T and ξ will have an impact on the time characteristics such as the charging times presented here. Even though absolute values will depend on the NVT parameters, intersystem comparisons using the same thermostat setup are still valuable. In all simulations of charging, the system was first equilibrated for 4 ns without applied potential before a production run.
Experiments
Synthesis of Novolacderived carbon
The synthesis of Novolacderived carbon involves three main steps^{57}. Briefly, we first crosslinked our polymer precursor using a solvothermal technique. 20 g of Novolac (ALNOVOL PN320, Allnex) was dissolved in 100 mL ethanol. Then, 2.5 g of crosslinker (hexamethylenetetramine) was dissolved in 500 mL miliQ water. The novolac solution was then added to the crosslinker solution in a 1 L autoclave container. At this step, we observed that the colour of the mixed solution changed from colourless to milky solution indicating the selfemulsion process. Before heating the autoclave, the container was filled with N_{2} gas to avoid oxidation of our polymer solution. Then, the autoclave was heated to 150 °C with the heating rate of 5 °C/min. The autoclave was held at 150 °C for 8 h and passively cooled down to room temperature. The asobtained sample was freezedried using liquid nitrogen to obtain novolacbeads. As for the second step, we pyrolyzed novolacbeads under an argon atmosphere at 700^{∘}C using a heating rate of 2^{∘}C/min. The pyrolysis time was 2 h. To enlarge the pore of carbon, as well as to enhance the surface area of carbon, the CO_{2} activation was necessary. The sample was subjected into the tubular furnace and heated to 1000^{∘}C, while feeding CO_{2} gas with the constant flow rate of 100 cm^{3}/min. The activation time was 2 h. The activated Novolacderived carbon was named PNC2h.
Material characterization
The electron micrograph of PNC2h was obtained via fieldemission scanning electron microscope (SEM; JEOLJSM7500F, JEOL Ltd). Nitrogen gas sorption analysis was performed with a Quantachrome Autosorb iQ system. Before nitrogen sorption, the sample was vacuum degassed at 200 °C for 1 h. After, the sample was heated to 300 °C and hold at such temperature for 20 h at the relative pressure of 0.1 Pa. Yet, the sample are volatile free. We then conduct the nitrogen sorption analysis at the temperature of −196 °C using liquid nitrogen. The relative pressure of the nitrogen was increased from 5 × 10^{−7} to 1.0 in 79 steps. A quenchedsolid density functional theory (QSDFT) was used to calculate the pore size distribution (PSD) assuming slitlike pores.
The scanning transmission electron microscope experiments were performed in the bight field mode on a JEOL Cs corrected ARM operated at 200 kV equipped with a cold fieldemission microscope with a nominal 0.1 nm probe size under standard operating conditions.
Electrode preparation and cell assembling
We used freestanding polymerbound electrode produced as described below^{57,75}. The carbon material PNC2h was first dispersed in ethanol. The mixture was stirred for 5 min, then polytetrafluorethylene (PTFE; 60 mass % in H_{2}O) was added into the carbonethanol mixture. The slurry was constantly mixed in the motor until the ethanol was evaporated. The doughlike carbon paste was then pressed into the squarish shape and rolled using the rolling machine to adjust the thickness of the electrode to 120 μm (wet thickness). The carbon electrode was then dried in the vacuum oven at 120 °C overnight. The resulting electrode consist of 90 mass % carbon and 10 mass % of PTFE. The dry thickness of our working electrode was ca. 100 μm.
We conducted the electrochemistry measurement using our custombuilt cell^{75}. We used fullcell symmetrical twoelectrode setup having PNC2h electrode as the working and counter electrode. First, PNC2h electrode was cut in a disk with a diameter of 12 mm. This electrode was attached to a graphitecoated aluminium current collector of the same diameter having the thickness of 37 μm. A 13 mm in diameter glass fibre (GF/F Whatman) was used as separator. After putting all the component into the body of the cell, the cell was tightly closed using springloaded titanium piston while leaving hold at the side of the cell open. The cell was then dried in the vacuum oven at 120 °C overnight to remove the residual humidity. Finally, the dried cell was filled by the ionic liquid (EMIMBF_{4}) in an MBraun Argonfilled glovebox (O_{2}, H_{2}O < 1 p.p.m.).
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We thank Jakob Krummacher for fruitful discussions and Cheng Lian for determining the prefactor α from the data of Fig. S9 of the Supplementary Material of ref. ^{58}. M.J. and S.K. acknowledge support from S. Dietrich (MPIIS, Stuttgart). V.P. and P.S. thank Eduard Arzt (INM) for his continued support. C.H. and K.B. were funded by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy  EXC 2075  390740016 and through Project Number 327154368 – SFB 1313.
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S.K., C.H. and K.B. initiated the research. K.B. performed the simulations. V.P. designed and supervised the experimental work, P.S. performed the experiments and B.L.M. did the TEM analysis. S.K. and M.J. derived the equations and drafted the manuscript. All authors contributed to the discussion of the results and editing of the manuscript.
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Breitsprecher, K., Janssen, M., Srimuk, P. et al. How to speed up ion transport in nanopores. Nat Commun 11, 6085 (2020). https://doi.org/10.1038/s41467020199036
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