Charging dynamics of an individual nanopore

Meso-porous electrodes (pore width « 1 µm) are a central component in electrochemical energy storage devices and related technologies, based on the capacitive nature of electric double-layers 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 nano-slits, 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 nano-slit 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.

Both measurements were taken in the so-called electric double-layer region, where the electrode is ideally polarized, indicating that neither oxidation not reduction occur at the gold surface. CV measurements in LiClO4 were taken using the three-electrode configuration in the SFB with the gold as a working electrode and platinum both as a counter and as a (quasi) reference electrode. Measurements in NaNO3 were taken using a similar three-electrode configuration using an external electrochemical cell.

Supplementary note 1: Estimate of effective gold surface potential
The potential applied at the gold surface at t = 0 by app, changes the equilibrium potential at the gold surface from gold1 to gold2; these equilibrium values can extracted from the normal force profiles (Figure 2 in main text). However, the instantaneous effective potential at the gold surface at t = 0 must depend both on gold1 and on gold2. The reason for this is as follows: At t < 0 the ion cloud in the vicinity of the gold surface exactly neutralizes the charge on the gold which is at a surface potential gold1. At t = 0 the gold surface charge instantaneously changes but the ion cloud has not yet had time to adjust to its equilibrium value (which happens only after a time EDL). Thus the effective potential eff at the gold surface at t = 0 arises from that due to the 'old' charge cloud (t < 0) in the vicinity of the gold surface, together with that due to the new charge on the gold surface induced by app. This is consistent with the shift D in separation between the surfaces on application of app, as shown in Supplementary Figure 1.
We see in Supplementary Figure 1 that the force between the surfaces (F = knD) is the same whether they go from gold1 to gold2 or from gold2 to gold1. This demonstrates that the force -and thus the effective potential at t = 0 -does not depend on either gold1 or gold2 by themselves, but on their combination. Since the potential at the surface arises in part from the 'old' ion charge cloud associated with gold1 and in part from the new surface charge associated with gold2, we may estimate the effective surface potential instantaneously following app at t = 0, as eff = (gold1 − gold2)/2. This is the value used in the expression for Fe in eq. (2) of the main text.

Supplementary note 2: Estimate of effective radius L of confined nano-pore within the transmission line model
The transmission line model for charging within a cylindrical nano-pore 1 considers a cylinder of length L and thickness hp, closed at one end and exposed to the bulk solution (i.e. a reservoir of ions) at its open end, so that L is the distance that ions from the reservoir need to diffuse to lead to EDL charging across the entire nanopore. Our present configuration (Figure 2b in main text) is a disk-like pore symmetric about its mid-plane, confined between two orthogonally-crossed cylinders, mean radius R (≈ 1 cm) and closest separation Di (≈ O(100 nm)). The effective geometry of such a configuration both for electrostatic 2 and hydrodynamic 3 interactions is that of a sphere (radius R) on a flat a closest distance Di apart (in the Derjaguin approximation 2 , which applies well for the present case of Di << R). The thickness of the disk varies between Di and (Di + (r 2 /2R)) at a distance r from the center.
We look for an effective diameter r corresponding to the length L of the cylinder in the TL model. Thus in our configuration L is the distance that ions from a region of sufficient ion excess (corresponding to the ion reservoir in the TL model) need to diffuse to the disk center. Consider a change in the potential of the gold which results in a change  in the surface charge density. This will lead to a change /unit area of ions that need to be compensated. We assume that ions need to travel to the disk center (r = 0) from a region at r = L where the excess number of ions in the gap relative to that at the disk center exceeds the number needing to be replaced. At r = L the excess gap thickness  relative to the center (r = 0) is  = (L 2 /2R), and the excess of ions relative to the disk center is then c0/unit area where c0 is the bulk ion concentration (number of ions per unit volume). We thus require c0e > , or L > (2R/c0e) 1/2 , where e is the electronic charge (assuming singly-charged ions). This is roughly the distance that ions need to travel to the disk center from a region (r = L) where there is sufficient ion excess to compensate for the charge density change at the disk center (r = 0) arising from the applied potential change app, so that it can act as a reservoir.

Supplementary note 3: EDL charging dynamics at different applied potentials
The surface charge density difference of the gold for the two potential steps shown in Parameters used to generate curves in Figure 4: Solid and dotted black curves in Figure 4a and upper black curve in Figure 4b, all in 2mM NaNO3, are from equation (3) with the same parameters as for Figure 3b above. Lower black curve in Figure 4b is from a different experiment, in 5 mM NaNO3, for which the parameters were determined separately: gold,eff = 0.084 V (determined as detailed in section (1)