Abstract
Mesoporous electrodes (pore width « 1 µm) are a central component in electrochemical energy storage devices and related technologies, based on the capacitive nature of electric doublelayers at their surfaces. This requires that such charging, limited by ion transport within the pores, is attained over the device operation time. Here we measure directly electric double layer charging within individual nanoslits, formed between gold and mica surfaces in a surface force balance, by monitoring transient surface forces in response to an applied electric potential. We find that the nanoslit charging time is of order 1 s (far slower than the time of order 3 × 10^{−2} s characteristic of charging an unconfined surface in our configuration), increasing at smaller slit thickness, and decreasing with solution ion concentration. The results enable us to examine critically the nanopore charging dynamics, and indicate how to probe such charging in different conditions and aqueous environments.
Introduction
Conducting materials with a large surfacetovolume ratio, such as mesoporous electrodes (pore widths « 1 μm), are important in a range of technologies^{1,2,3,4,5,6}. Prominent among these are supercapacitors^{4,5}, but also batteries^{7,8}, fuel cells^{9,10} and electrocatalysts^{11,12}, while emerging applications include capacitive deionization^{13,14}, and the extraction of renewable energy through capacitive mixing of aqueous solutions of different salt concentrations^{15,16}. Charge is stored in supercapacitors when electrolyte ions form (capacitive) electrical double layers (EDLs) at the surface of oppositely charged electrodes under an externally applied voltage^{17}. Since the amount of charge stored is proportional to the available electrode surface area, materials with a high specific area (such as porous electrodes) are clearly at a premium; but they are only fully useful if ion transport enables equilibrium EDL formation over the entire pore within the relevant charging/discharging time. It is crucial therefore to be able to characterize and understand the dynamics of EDL formation within nanopores.
In equilibrium, the EDL at a charged surface screens the associated electric field, with an exponential decay length (the Debye length) \(\lambda _{\mathrm{D}} = \sqrt {\frac{{\varepsilon \varepsilon _0k_{\mathrm{B}}T}}{{2c_oe^2}}}\) (for 1:1 electrolyte), where c_{0} is the concentration of salt in the bulk solution, ε_{0} is the permittivity of free space, ε is the dielectric constant, k_{B}T the thermal energy, and e the electronic charge^{18}. Changes in the surface potential rearrange the ionic atmosphere comprising the EDL to a new equilibrium configuration^{19}, a dynamic process known as EDL charging. This is well understood for planar electrodes in bulk solution, where the characteristic charging time is defined by τ_{c} = λ_{D}H / Ɗ (where H is the distance between electrodes and Ɗ is the ion diffusion coefficient)^{19,20,21,22}, but for the case of EDL charging in nanoconfined geometries (as in porous electrodes) the dynamics may be much slower. This is because, in small diameter pores, changes to the EDL are constrained by the limited supply of ions within the pore^{23,24,25,26,27} and the time for ions to access the entire pore from the reservoir with which it is in contact.
A widely used description of ion transport in pores is the classic “TransmissionLine” (TL) model, by de Levie^{6,24,28}, where the pore – a closed cylinder of length L_{p} and diameter h_{p} (L_{p} » h_{p} » λ_{D}) in contact with an ion reservoir  is treated as an equivalent TL circuit composed of a set of resistors and capacitors, yielding a time for EDL charging within the pore τ_{TL} = (L_{p}^{2} / Ɗ)(λ_{D} /h_{p}). This characteristic time scales as the diffusion time (L_{p}^{2}/Ɗ) for ions to traverse the pore length L_{p}, modulated by the ratio (λ_{D}/ h_{p}). The latter is the extent of the nearsurface EDL region (~λ_{D}), within which ion redistribution must occur due to the potential change, relative to the pore diameter h_{p}, within which little overall change occurs. The charging dynamics of an EDL confined in a single nanopore have been investigated primarily theoretically, e.g. refs. ^{6,29,30,31,32,33,34,35}, while several treatments have extended the TL treatment, and different charging dynamics have recently been predicted for EDLs confined in dimensions comparable with its Debye length^{6,28,36}. Experimentally, charging dynamics have been probed within (macroscopic) porous electrodes^{25,37,38,39}, but, to our knowledge, no measurements of EDL charging dynamics within a single nanopore – the basic element of such electrodes—have been reported to date.
Here we use a surface force balance (SFB) to probe experimentally the charging dynamics of an EDL confined to a nanoslit between two surfaces, in response to a stepchange in the surface potential (or charge) of one of them (arising from a step ∆Ψ_{app} in the potential applied to it). We find that the response to such a step is governed by two successive processes: Immediate formation of a strong electric field across the pore due to elimination of the screening, arising from the instantaneous charge imbalance in the EDL, followed by a slower process in which the ionic atmosphere of the EDL rebuilds to screen the generated electric field while charging the nanopore interior (the gold acquires its new surface potential value over a time δt much faster than EDL charging time as it is associated with drift motion of electrons which is orders of magnitude faster than for ions in solution^{11,40}). Our results show that full EDL charging within the nanoslit occurs over times of order 0.5–1 s, far longer than τ_{c} ≈ 3 × 10^{−2} s in our configuration (where τ_{c} = λ_{D}H / Ɗ, and the values of λ_{D}, H and Ɗ are given below). They reveal that the charging time increases at smaller slit thickness, and decreases with increasing ion concentration in the electrolyte solution, suggesting that the process of EDL charging within the nanoslit is largely limited by diffusion of ions from the bulk reservoir to which the nanoslit is coupled^{24,28,36}. These findings demonstrate the ability of the SFB to access and probe the EDL charging dynamics within a single nanoscale gap (« 1 μm), the basic element of porous electrodes, providing a method for probing EDL charging in nanoconfined slits with different surfaces and under different conditions.
Results
Probing EDL charging in a nanoslit with the SFB
Figure 1a shows a schematic of the experimental SFB configuration used in this study, where a singlecrystal mica surface faces a smooth gold surface (r.m.s. roughness ca. 3 Å) at a controlled potential^{41,42,43}. The absolute separation D between the surfaces is determined (to ± 3 Å) by multiple beam interference and monitored via videorecording of the wavelengths of the interference fringes^{44,45}.
We probe the perturbation and the charging of the EDL through its transient effect on the force between the gold and mica surfaces. A typical dynamic measurement, conducted in 5 mM NaNO_{3} (λ_{D} = 4.3 nm), is shown in Fig. 1c. Initially, the surfaces are brought to a given initial separation D_{i} (» λ_{D}). Then, either a positive (–0.2 V → + 0.2 V) or negative ( + 0.2 V → −0.2 V) potential step ∆Ψ_{app} is applied to the gold while monitoring the goldmica separation D (Fig. 1c). After ∆Ψ_{app} is applied at time t = 0 (Fig. 1d), the gold electrode – both outside and within the confined nanoslit – accumulates a new surface charge density over a very short time δt (typically < 10^{–9} s due to drift motion of electrons^{11,40}) as it attains its new surface potential value Ψ_{gold}. This generates (at δt) an unscreened electric field^{19} which exerts an electrostatic force F_{e}(t = δt ≈ 0) on the negatively charged mica, bending the spring K_{n} on which it is mounted by ∆D to an extremal separation D_{f} relative to the gold, over a time ∆t_{s}. As ions transport to reconstruct the EDL (EDL charging) within the nanopore, F_{e} is progressively screened and the surfaces relax to their initial separation D_{i} over a further time ∆t_{r} (Fig. 1d), since, in all cases in our study, D_{i} » λ_{D}. This transient motion manifests as a peak, with a characteristic asymmetric shape, in the D vs. t trace (Fig. 1d) and a period of (∆t_{s} + ∆t_{r}) which characterizes the charging time of the EDL within the nanoslit. Away from the nanoslit – at the unconfined gold surface  the EDL equilibrates over a time τ_{c}^{19} ( = λ_{D}H/Ɗ ≈ 3 × 10^{−2}s), which is much shorter than the EDL charging time (0.5–2 s) within the nanopore measured in our experiments. For example, for the configuration of Fig. 1d, a characteristic time (∆t_{s} + ∆t_{r}) ≈ 0.85 s was obtained in response to a potential step of 0.2 V → + 0.2 V. Overall, this indicates three different time scales following the potential step: (∆t_{s} + ∆t_{r}) ≫ τ_{c} ≫ δt. Thus, the times (∆t_{s} + ∆t_{r}) we measure in our experiments, corresponding to the EDL equilibration within the nanopore, are well separated from τ_{c}, the time for EDL equilibration at the unconfined gold away from the nanopore.
The initial (t = 0) instantaneous electric field across the gap depends on the effective surface potentials Ψ_{gold} at the gold surface at different applied potentials Ψ_{app}; these may be extracted from normal force vs. surfaceseparation D profiles, as shown in Fig. 2a. In Fig. 2b are shown D(t) traces in response to Ψ_{app} = –0.2 V → + 0.2 V and back again, at selected D_{i} values covering the range of pore widths D_{i} examined. Importantly, this also demonstrates the repeatability and reproducibility of our dynamic measurements, indicating that no chemical reaction occurs at the gold (this is shown directly by cyclic voltammetry, ref. ^{45} and in the Supplementary Figure 4). In Fig. 2c are magnified peaks at different D_{i} values. Such transient peaks reflect the EDL perturbation by ∆Ψ_{app} (at t = 0) and its subsequent charging, and are analyzed below (for clarity we focus on the –0.2 V → + 0.2 V transition though identical considerations apply to the reverse transition).
Transient forces during EDL charging
The force F_{e}(t) attracts the charged mica surface due to the induced positive charge on the gold surface, changing the surface separation D_{i} by ∆D(t) = (D_{i} – D(t)). This motion is opposed by the bending of the spring (of constant K_{n}) on which the mica surface is mounted, which exerts an opposing force F_{k} = K_{n}∆D, and by a hydrodynamic damping force F_{H} (arising from extrusion of the liquid as the surfaces approach or separate). Van der Waals (vdW) attraction between the surfaces is negligible since D is always > 20 nm, and we may set the normal surface force as equal to F_{e}. The equation of motion is thus F_{e}(t) = F_{k} + F_{H} + m(d^{2}D/dt^{2}), where the last term is inertial and m is the mass of the mica surface and its mount. We note that the magnitude of m(d^{2}D/dt^{2}) (where m ≈ 3 × 10^{–3} kg), as determined from the D vs. t plots (Fig. 1d or Fig. 2b), is ca. 10^{–9}N, which is negligible compared with the hydrodynamic (F_{H} = O(10^{−6} N)) and spring (F_{k} = O(10^{–6} N)) force terms, and may be ignored. The hydrodynamic force, between a sphere approaching a flat a closest distance D away across a liquid of viscosity η, is^{46} F_{H} = 6πR^{2}η[(dD / dt) / D], and the equation of motion becomes:
Before solving equation 1, we address an important qualitative point concerning the EDL charging. Following the potential change on the gold surface at t = 0, F_{e} is a maximum but immediately begins to decay as the EDL recharges. During the initial motion (t < ∆t_{s}) F_{e} exceeds F_{k}, but at t = ∆t_{s}, where D = D_{f} (Fig. 1d), it has decayed so that F_{e}(∆t_{s}) = F_{k} = K_{n}(D_{i} – D_{f}). At longer times t > ∆t_{s}, F_{e} becomes smaller than the spring force F_{k}, and the mica surface is pushed back towards its equilibrium position D = D_{i} over the period ∆t_{r}. The crucial point is whether the period ∆t_{r} is due to hydrodynamic damping alone, or to a slower process due to F_{e} decay arising from increased screening associated with EDL charging limited by ion transport within the gap. To resolve this we solve for the motion of the surfaces from D = D_{f} at the peak to D = D_{i}, on the assumption of hydrodynamic damping alone. The relevant equation is simply equation (1) with F_{e} set to zero, and boundary condition D = D_{f} at t = 0. The resulting D(t) variation is given in Fig. 3 and shown as the red curves in Fig. 3a. We see that for traces at D_{i} ≳ 80 nm, D approaches D_{i} significantly slower than can be accounted for by hydrodynamic damping alone. This shows that D(t) is dominated by the slow decay in F_{e} rather than by hydrodynamic damping, so that (∆t_{s} + ∆t_{r}) = τ_{EDL} provides a measure of the time τ_{EDL} for the EDL charging (we note that for both hydrodynamic damping and EDL charging, the characteristic times vary as (1/D); however, at small D_{i} the mean value of D across the pore is significantly larger than D_{i}, which reduces the EDL charging time but leaves the hydrodynamic damping time unchanged, so that it becomes more dominant at the smallest D_{i} values such as D_{i} < ca. 80 nm. This is also manifested in the ∆D vs. D_{i} plot (Supplementary Figure 1) as a decrease in ∆D at D < ca. 80 nm). To solve equation (1), we require an explicit form for F_{e}(t). We approximate the instantaneous initial force (just after application of the potential change on the gold surface) as F_{e}(0) = πRεε_{0}(∆Ψ_{eff})^{2} / D(t = 0), which is the force between a conducting sphere (radius R) and a conducting plane a closest distance D apart^{47} (D « R) differing in potentials by ∆Ψ_{eff}. In our configuration ∆Ψ_{eff} = Ψ_{gold,eff} – Ψ_{mica} (at t = 0), the effective potential difference at t = 0 between the opposing gold and mica surfaces, where Ψ_{gold,eff} is the instantaneous effective gold surface potential (t = 0) arising from the abrupt variation of the gold surface charge (Supplementary Note 1). This may be evaluated from the potentials extracted from the force vs. distance profile at the relevant applied potentials^{44}, Fig. 2a. The decay with time of F_{e}(t) is complex, depending on flux of ions between the bulk and the nanoslit which in turn modifies the initial gold surface charge (i.e. at t = δt) within the confined area until the EDL is fully relaxed. To proceed, we may assume that F_{e} decays exponentially with a characteristic time τ due to the progressive screening of the electrostatic force with time, as the nearsurface ion concentration rearranges following the potential change. Equation (1) then becomes:
We seek an approximate solution by recognizing that D does not change too greatly from D_{i} during the transient response (particularly for the larger D_{i} values), and replace D by D_{i} in the denominators of the terms for F_{e} and F_{H}. This gives:
which for boundary condition D = D_{i} at t = 0, solves as:
where F_{e}(0) = πRεε_{0}∆Ψ_{0}(∆Ψ_{eff})^{2} / D_{i} and f_{H} = 6πR^{2}η / D_{i}.
Equation (3) describes the transient change in D in response to ∆Ψ_{app}. We note that at the peak in D(t), at t = ∆t_{s}, (dD/dt) = 0 so that for this value of t ( = ∆t_{s}), equation (2’) yields the value of τ directly. This may then be used in the expression for D (equation (3)). In Fig. 3b, this is done for the D vs. t traces at different D_{i} values, showing a close quantitative fit with no adjustable parameters (all parameters are determined, Fig. 3). We confirmed the validity of our approximate solution by numerically solving equation (2) (Supplementary note 4). The numerical solution shows the behavior of D(t) to be close to that predicted by equation (3) over the range of D values in our experiments, and as expected, closest at higher D values where the approximation leading to equation (3) is best (Supplementary Figure 3).
Effect of confinement on EDL charging time
In Fig. 4 we gather together results for the EDL charging times for different D_{i} values, that is, different thicknesses of the nanoslit, showing the variation with D_{i} of ∆t_{s}, ∆t_{r} and (∆t_{s +} ∆t_{r}) (inset to Fig. 4a). While the values shown are for ∆Ψ_{app} = −0.2 V → + 0.2 V and + 0.2 V →−0.2 V in 2 mM salt, similar behavior is seen at different salt concentrations (Fig. 4b), and for different salts (Fig. 4c), as well as for other potential variations (Supplementary note 3). In Fig. 4a the broken and solid black curves are the calculated values of ∆t_{s} and ∆t_{r}, respectively, from equation (3). The red curve in Fig. 4a is the calculated hydrodynamic damping time from the peak maximum (corresponding to the red curves shown in Fig. 3a), in the absence of any electrostatic force F_{e}, and shows that such damping alone cannot account for the significantly longer decay times ∆t_{r} at D_{i} ≳ 80 nm.
Figure 4b shows the variation of the overall charging time ((∆t_{s}(D_{i}) + ∆t_{r}(D_{i})) with D_{i} for two different salt concentrations, 2 mM (squares) and 5 mM (triangles), where the solid black curves are the calculated fits from equation (3) for the two cases. These curves show good agreement with the respective measured charging times (∆t_{s}(D_{i}) + ∆t_{r}(D_{i})), indicating that our equations, which provide a description of the dynamics, though they do not clarify the mechanism of the nanoslit charging, can provide a good description of the data. The blue curve is the variation of the characteristic charging time τ_{TL} according to the transmission line model for nanopore charging, described earlier and discussed in the following section.
Figure 4c shows the variation of the overall charging time when the NaNO_{3} salt in Fig. 4a, b is replaced by a different salt, LiClO_{4}, which moreover has much larger differences between its ionic diffusion coefficients (Li^{+} and ClO_{4}^{−}), as considered in the following section.
Mechanism of nanoslit charging
It is of interest to consider the EDL charging dynamics in the context of the classic transmission line (TL) approach noted earlier^{24,28,36}. In our configuration (Fig. 1b), the nanopore or nanoslit is the goldmica gap about the region of closest approach, D = D_{i}, strictly the region between two crossed cylinders of mean radius R. Its effective geometry – as in the Derjaguin approximation^{18}, which is valid for D_{i} « R in the present case – is that of a circular slit bounded by a spherical surface and a flat, with a slit radius r = L, in contact with an ion reservoir (because the slit is formed by two orthogonal cylindrical surfaces, it is symmetric about its midplane). L is the distance required by ions from the reservoir (at r = L) to fully permeate the pore to its center (r = 0), where its thickness is h_{p} = D_{i}, to compensate for the charge induced on the gold by the potential step ∆Ψ_{app}. Within the TL model, the characteristic charging time of the EDL within such a circular slit is then expected to scale as τ_{TL} = (L^{2} / Ɗ)(λ_{D}/h_{p})^{6,24}. The surface separation at r = L, D = D_{i} + (L^{2}/2 R) (Fig. 1b), must be sufficiently large to provide for the required excess of ions. This corresponds to a value L ≈ (2 R∆σ / c_{0}e)^{1/2} (Supplementary note 2), where ∆σ is the change in the gold surface charge density σ arising from ∆Ψ_{app}. In Fig. 4b, it is shown as a solid blue curve, the characteristic time τ_{TL} = (L^{2} / Ɗ)(λ_{D} / h_{p}) expected for a pore of length L, and width h_{p} = D_{i}. The fit to the experimental data corresponds to L = 150 μm. This is within a factor 2 of the value L = (2 R∆σ / c_{0}e)^{1/2} ≈ 87 μm evaluated for the corresponding potential step (∆Ψ_{app} = + 0.2 V → –0.2 V in 2 mM salt) (Supplementary Note 2), which is believed to arise from the different geometries considered in the two cases.
The characteristic time τ_{TL} = (L^{2} / Ɗ)(λ_{D} / h_{p}) also predicts that EDL charging is faster for higher salt concentrations, since (in our configuration) a larger c_{0} is associated both with a smaller L (~c_{0}^{–1/2}), as well as a smaller λ_{D} (~c_{0}^{–1/2}), suggesting τ_{TL} ~ c_{0}^{–3/2}. This is indeed observed for EDL charging from higher salt concentrations as shown in Fig. 4b, where the experimental charging time is roughly 3.5fold larger in the 2 mM NaNO_{3} salt compared with the 5 mM salt, similar to the expected ratio (c_{0, 2 mM}/c_{0, 5 mM})^{–3/2} ≈ 4.
In addition, we would expect from the expression for τ_{TL} that ions with different diffusion coefficients Ɗ would result in different charging times. Additional cations need to permeate the slit from the reservoir when the applied potential change renders the surface charge more negative (e.g. ∆Ψ_{app} = + 0.2 V → –0.2 V), and additional anions need to permeate for the opposite case. For the NaNO_{3} salt, Fig. 4a, b, where the Ɗ values for Na^{+} (Ɗ = 1.3 × 10^{–9} m^{2} s^{–1}) and NO_{3}^{–}(Ɗ = 1.9 × 10^{–9} m^{2} s^{–1}) differ by some 40%, there is a slight indication that EDL charging is slower when ∆Ψ_{app} = + 0.2 V → – 0.2 V (red squares in Fig. 4a, when it is controlled by transport of slower Na^{+} ions) relative to ∆Ψ_{app} = – 0.2 V → + 0.2 V (green squares, when charging occurs by transport of NO_{3}^{–} ions). However, when NaNO_{3} is replaced by LiClO_{4}, where the Ɗ values for the latter (1 × 10^{–9} m^{2} s^{–1} and 1.8 × 10^{–9} m^{2} s^{–1} for the Li^{+} and ClO_{4}^{–} ions respectively) differ by twice as much as for the former, there is a more marked effect of the mobility of the ion type permeating the slit on the charging time. This is seen by the differences between the red (charging by Li^{+} transport) and green (charging by ClO_{4}^{−} transport) data in Fig. 4c. This is especially the case in Fig. 4c for D_{i} > ca. 80 nm, where ion transport rather than hydrodynamic effects dominates the dynamics (see Figs. 3a and red curve in 4a); indeed, the difference between the charging times for anionic vs. cationic transport demonstrates clearly that the decay of the transient forces that we measure is dictated by diffusivetransport of ions from the reservoir. Finally, we would expect that different potential steps would result in a different charging times through their effect on the surface charge density change ∆σ and hence on the effective slit radius L (~ (∆σ)^{1/2}), as indeed is seen (Supplementary Figure 2). Thus, this broad agreement of the absolute magnitude as well as the variation of the predicted charging times τ_{TL} = (L^{2} / Ɗ)(λ_{D} / h_{p}) with our measured EDL charging for different slit widths, salt concentrations, ionic mobility and surface potential changes, indicates the validity of the TL concept and its scaled behavior for our system.
Discussion
In summary, we report the first measurements of EDL charging dynamics within a single nanoconfined circular pore (or slit) following a step change in the surface potential (and charge) of one of the confining surfaces. This is done, using an SFB, by monitoring the transient change in surface forces due to an instantaneous electric field arising from the EDL perturbation, which decays as the EDL charges. Our results are in line with the characteristic times expected from the transmission line model applied to our nanoslit geometry, and indicate that such an approach could be used for probing EDL charging dynamics within single nanoslits in different conditions and aqueous environments.
Methods
Materials
Gold pellets, 99.999% pure, were purchased from Kurt J. Lesker and evaporated from a graphite crucible. Sodium nitrate, NaNO_{3}, 99.99% pure was purchased from Merck Millipore and used as received. Lithium perchlorate, LiClO_{4}, 99.99% pure was purchased from SigmaAldrich and used as received.
Preparation of salt solutions
NaNO_{3} and LiClO_{4} were dissolved in purified water with a total organic content of less than 1 ppb (TOC < 1), a resistivity of 18.2 MΩ cm and pH = 5.8.
Surface force balance experiment
An atomically smooth back silvered mica and molecularly smooth gold surface are glued to cylindrical fusedsilica lenses (curvature radius R ≈ 1 cm) in a crossedcylinder configuration, equivalent to the geometry of a sphere on a flat, with the lower lens (mica) mounted on a horizontal leaf spring (spring constant K = 81.5 ± 2.7 Nm^{−1}) and the top lens (gold) mounted on a sectored piezoelectric tube (PZT). The gold surface, serving as a working electrode (W), is connected to a potentiostat together with two platinum wires, acting as reference (R) and counter (C) electrodes, as described in detail elsewhere^{44,45}. During all measurements, the electrodes are immersed in an electrolyte solution inside a quartz bath, customdesigned to prevent leakage of current to the ground, and the distance H between them is of order a few mm. The distance between the surfaces is measured with a resolution of 1.0 ± 0.3 nm using multiple beam optical interferometry, and the normal forces between the surfaces are monitored through the bending of the horizontal spring and measured via a dynamic approach with a force sensitivity of ~50–100 nN, as previously described in detail^{48,49}.
Video capture and analysis
The absolute surface separation between the surfaces during EDL charging and dynamic force measurements was continuously monitored, by capturing the fringes of equal chromatic order (FECO) position, using a fast video recording camera (SONY XCHR70) with a frame rate of 60 frames per s. In our gold/medium/mica/silver twolayer interferometer system each pixel is equivalent to an absolute distance of ca. 0.2–0.3 nm. The separation between the surfaces was determined using the multilayer matrix method, as previously described in detail^{50,51}.
Data availability:
The data sets generated and analyzed during the current study are available from the corresponding author on reasonable request.
Change history
02 May 2019
The original version of this Article contained an error throughout in which an incorrect symbol was used for the diffusion coefficient: it should be cambria math, italicized, and not bold. These have been corrected in both the PDF and HTML versions of the Article.
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Acknowledgements
We thank Ohad Cohen for his help with the numerical solutions, Alexander Vaskevich for helpful discussions and Maarten Biesheuvel for useful comments. We thank the European Research Council (Advanced grant CartiLube), the Israel Science Foundation and the Minerva Foundation for financial support. This research was made possible in part by the historic generosity of the Harold Perlman family.
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J.K., R.T. and G.S. conceived the idea and designed the experiments. R.T. conducted the experiments and analyzed the data. S.S. and P.P. assisted with theoretical aspects of EDL charging dynamics. R.T. and J.K. wrote the manuscript. All authors discussed the results and commented on the manuscript.
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Correspondence to Jacob Klein.
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Tivony, R., Safran, S., Pincus, P. et al. Charging dynamics of an individual nanopore. Nat Commun 9, 4203 (2018). https://doi.org/10.1038/s41467018063641
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