Diffusion Limitations and Translocation Barriers in Atomically Thin Biomimetic Pores

Ionic transport in nano- to sub-nano-scale pores is highly dependent on translocation barriers and potential wells. These features in the free-energy landscape are primarily the result of ion dehydration and electrostatic interactions. For pores in atomically thin membranes, such as graphene, other factors come into play. Ion dynamics both inside and outside the geometric volume of the pore can be critical in determining the transport properties of the channel due to several commensurate length scales, such as the effective membrane thickness, radii of the first and the second hydration layers, pore radius, and Debye length. In particular, for biomimetic pores, such as the graphene crown ether we examine here, there are regimes where transport is highly sensitive to the pore size due to the interplay of dehydration and interaction with pore charge. Picometer changes in the size, e.g., due to a minute strain, can lead to a large change in conductance. Outside of these regimes, the small pore size itself gives a large resistance, even when electrostatic factors and dehydration compensate each other to give a relatively flat—e.g., near barrierless—free energy landscape. The permeability, though, can still be large and ions will translocate rapidly after they arrive within the capture radius of the pore. This, in turn, leads to diffusion and drift effects dominating the conductance. The current thus plateaus and becomes effectively independent of pore-free energy characteristics. Measurement of this effect will give an estimate of the magnitude of kinetically limiting features, and experimentally constrain the local electromechanical conditions.

. IV-characteristics. K + current through graphene crown ether pore at different strains in 1 mol/L KCl versus voltage. When current is limited by barriers, the IV-characteristics are super-linear and, when current is limited by diffusion, the IV-characteristics are sub-linear. Linear behavior may appear due to the cancellation of the two effects. While some of the data fit well to a simple power law, there are prominent features due to the rather complex dependence of energetics on voltage. The error bars are plus/minus one SE from five parallel runs.
Golden aspect ratio: The role of the bulk in determining resistance-specifically in access or diffusion limitations-requires a careful treatment of the simulation cell, as the bulk only slowly converges. We thus employ the golden aspect ratio method [2,3]. This employs a special aspect ratio that, after the disappearance of non-scaling finite-size effects, converges immediately to the infinite bulk limit. Figure S2 shows the current versus the simulation cell cross-sectional length (with height proportional to this length according to the golden aspect ratio). While there is some variation in the current over the length scales shown, it is mostly within the statistical error bars (from five parallel runs). Some of this variation may be non-scaling finite size effects, but overall the change is within expected errors.
. Potassium current through the crown ether graphene pore with various simulation cell cross-sectional lengths but with the same aspect ratio (H/L ≈ 1.2) at q O = −0.54 e and strain 6 %, showing that the ionic current does not vary with simulation cell size as long as the aspect ratio is kept at the golden aspect ratio. The error bars are plus/minus one SE from five parallel runs.

Concentration effects:
In the main text, we discuss how for q O = −0.54 e at small strain and low voltage the current displays many-body blockade effects. Otherwise, the other parameter regimes are single-ion transport. One way to see many-body effects is to study the concentration dependence of the current. Figure S3 shows the normalized current versus the concentration. For q O = −0.54 e at small strain, this normalized current decreases versus concentration, which suggests that many-body effects are at play (this dependence is just outside the error bars). Increasing the concentration does not increase the current proportionally since the main pore site is already occupied and is preventing further current flow. All other cases show only small variations or small upward trend, albeit barely out of the range of the statistical error bars from the five parallel runs. Occupancy: Within the main text, we develop a three site model for the current through the q O = −0.54 e pore. The form of this model is motivated by concentration and rate data. However, it still has several parameters. To reduce the number of parameters, we compute the occupancy of the main pore site versus strain and voltage. As an input, this allows a more rigorous treatment of the fitting of the MD current data to the model. Figure S4 shows the potassium occupancy of the main pore site versus strain for the four voltages we study. The occupancy in the q O = −0.24 e pore does not (except at low voltage) display any well-defined trends. At low voltage and within the main pore region, the free energy is increasing (the satellite barriers are decreasing), giving rise to an exponential decrease in pore occupancy. For other voltages, there is a more complex interplay of strain and voltage within the free energy landscape. However, for q O = −0.54 e, there is a clear exponential decrease in the pore occupancy with strain due to the raising of the bottom of the potential well. Moreover, the occupancy decreases a bit faster than exponential with voltage (in the main text, we model the dissociation time as Ve vV , with v some positive constant, which is both physically motivated-the exponential represents a decrease in barrier height and the pre-exponential the local field driving the ions-and works well).
Here, we also show additional concentration data versus spatial position, see Figure S5 and Figure S6. This shows where the concentration can increase or decrease with voltage depending on the pore characteristics. It also motivates the three site model we take in the main text. Free energies and electrostatic potentials: Figures S7 and S8 show additional free energy profiles, as well as a larger range of z. These demonstrate that there are indeed irrelevant features in free energy. Specifically, q O = −0.54 e has a feature at 0.2 nm for most strains that changes little when the voltage is taken from 0 V to 0.25 V. Moreover, the free energy is near barrierless for the largest strain examined. For q O = −0.24 e, the barrier in the middle of the pore increases with strain until somewhere between 4 % and 6 % strain, after which the peak mostly broadens and then decreases. This is due to the decrease of the electrostatic interaction initially being unable to compensate for dehydration. At 0.25 V, the pore is nearly barrierless at high strain, as the bias effectively wipes out the main feature.  Other: In the remaining figures of the SM, we show additional one-way rate data. Figures S11 and S12 show the one-way rate data as in the main text (across z-planes) for the rest of the parameter regimes. Figure S13 instead shows one-way rate data across hemispherical surfaces.
10.0% The inward diffusion rate of K + at a distance 0.6 nm, 0.4 nm, and 0.3 nm from the pore versus applied voltage. The dashed horizontal lines give the rate from the diffusion equation assuming ions only enter through a quarter of the sphere (i.e., π D c r). For small voltage, the k in near the pore is much smaller than diffusion limit and thus the current is limited by the barrier to transport. At large voltage, k in , and hence the current, approach the diffusion limit. The error bars are plus/minus one SE from five parallel runs.