Nanoscale electrohydrodynamic ion transport: Influences of channel geometry and polarization-induced surface charges

Electrohydrodynamic ion transport has been studied in nanotubes, nanoslits, and nanopores to mimic the advanced functionalities of biological ion channels. However, probing how the intricate interplay between the electrical and mechanical interactions affects ion conduction in asymmetric nanoconduits presents further obstacles. Here, ion transport across a conical nanopore embedded in a polarizable membrane under an electric field and pressure is analyzed by numerically solving a continuum model based on the Poisson, Nernst-Planck, and Navier-Stokes equations. We report an anomalous ionic current depletion, of up to 75%, and an unexpected rise in current rectification when pressure is exerted along the external electric field. Membrane polarization is revealed as the prerequisite to obtain this previously undetected electrohydrodynamic coupling. The electric field induces large surface charges at the pore tip due to its conical shape, creating nonuniform electrical double layers (EDL) with a massive accumulation of electrolyte ions near the orifice. Once applied, the pressure distorts the quasiequilibrium distribution of the EDL ions to influence the nanopore conductivity. Our fundamental approach to inspect the effect of pressure on the channel EDL (and thus ionic conductance) in contrast to its effect on the current arising from the hydrodynamic streaming of ions further explains the pressure-sensitive ion transport in different nanochannels and physical regimes manifested in past experiments, including the hitherto inexplicit mechanism behind the mechanically activated ion transport in carbon nanotubes. This enhances our broad understanding of nanoscale electrohydrodynamic ion transport, yielding a platform to build nanofluidic devices and ionic circuits with more robust and tunable responses to electrical and mechanical stimuli.

Figure 1

(a) A cross section of a conical nanopore embedded in a dielectric membrane separating two KCl electrolytic reservoirs is shown in the cylindrical coordinates (r, z). A voltage $\mathrm{\Delta }V$ and pressure $\mathrm{\Delta }P$ are applied in the negative-$z$ direction. (b) A chart illustrates the equations considered to study the ion transport across the nanopore, their dependent variables, and the exchange of the variables needed between them. At $\mathrm{\Delta }P=0$, $\sigma =-0.03\mathrm{C}/{\mathrm{m}}^2$, $\varepsilon =12$, $R_{\mathrm{t}}=8$ nm, $h=55$ nm, $\theta ={74}^{\circ }$, and $c_0=35$ mM, (c) and (d) display 2D contours of the net ionic concentration ($c_{+}-c_{-}$) for $\mathrm{\Delta }V=4$ V and −4 V, respectively, while (e) evaluates the distribution of induced surface charge density (${\sigma }_I$) on the nanopore sidewall (AH) using Eq. (4b).

Figure 2

Current-voltage (I-V) plots for the conical nanopore at positive and negative pressures ($\mathrm{\Delta }P$) are shown in (a) and (b), respectively, while (c) and (d) illustrate the current-pressure (I-P) curves at positive and negative voltages ($\mathrm{\Delta }V$), respectively. The simulation parameters used here are the same as in Fig. 1.

Figure 3

Total ionic concentrations ($c_{+}+c_{-}$) and axial electric field intensities $\left(E_z\right)$ on MN, a line near and parallel to the lateral nanopore surface [see the inset of (a)], are plotted in the left [(a) and (b)] and right [(c) and (d)] panels, respectively, at different pressures ($\mathrm{\Delta }P$). The applied voltages ($\mathrm{\Delta }V$) in the top [(a) and (c)] and bottom [(b) and (d)] panels are 4 and −4 V, respectively. The 2D contours in the insets of (c) illustrate the electric field intensities in the negative-$z$ direction at $\mathrm{\Delta }V=4$ V as $\mathrm{\Delta }P$ varies from −6 to 6 MPa. The profiles of electric potential and the electric field intensity in the positive-$z$ direction at $\mathrm{\Delta }V=-4$ V in the insets: (d)i and (d)ii, respectively, undergo trivial changes with $\mathrm{\Delta }P$. The simulation parameters used are the same as in Fig. 1.

Figure 4

Two-dimensional contours of the total ionic concentration ($c_{+}+c_{-}$) across the conical nanopore are plotted using the simulation parameters in Fig. 1. The applied pressure ($\mathrm{\Delta }P$) varies from −6 to 6 MPa from left to right, while the applied voltage ($\mathrm{\Delta }V$) in the top and bottom rows are 4 and −4 V, respectively.

Figure 5

The effects of the nanopore tip radius ($R_{\mathrm{t}}$) on the maximum depletion factor ($\mathrm{D}{\mathrm{F}}_{\max }$) and the optimal pressure ($P_{\mathrm{opt}}$) are shown in (a) and (b), respectively, for $\sigma =-0.02\mathrm{C}/{\mathrm{m}}^2$ and $h=55$ nm. The applied voltage ($\mathrm{\Delta }V$) and membrane dielectric constant ($\varepsilon$) are simultaneously varied to demonstrate the influence of polarization-induced surface charges. We then plot the effects of the (negative) inherent surface charge density magnitude ($\sigma$) on $\mathrm{D}{\mathrm{F}}_{\max }$ and $P_{\mathrm{opt}}$ in (c) and (d), respectively, for $h=55$ nm, $R_{\mathrm{t}}=8$ nm, $\mathrm{\Delta }V=5$ V, and varying $\varepsilon$. We further illustrate the variations of $\mathrm{D}{\mathrm{F}}_{\max }$ and $P_{\mathrm{opt}}$ with the pore length ($h$) in (e) and (f), respectively, for $\sigma =-0.03\mathrm{C}/{\mathrm{m}}^2$, $R_{\mathrm{t}}=8$ nm, $\mathrm{\Delta }V=4$ V, and different values of $\varepsilon$. We fix θ at $54.7^{\circ }$ and $c_0$ at 100 mM in all the plots of (a)–(f).

Figure 6

The effects of the complementary cone angle (θ) on the maximum depletion factor ($\mathrm{D}{\mathrm{F}}_{\max }$) and the optimal pressure ($P_{\mathrm{opt}}$) are plotted in (a) and (b), respectively, for $c_0=100$ mM. The variations of $\mathrm{D}{\mathrm{F}}_{\max }$ and $P_{\mathrm{opt}}$ with the reservoir salt concentration ($c_0$) are displayed in (c) and (d), respectively, for $\theta ={74}^{\circ }$. In both sets of plots, we take $\sigma =-0.03\mathrm{C}/{\mathrm{m}}^2$, $\varepsilon =12$, $R_{\mathrm{t}}=8$ nm, $h=55$ nm, and $\mathrm{\Delta }V=5$ V. The (orange) shaded data points in (a) and (c) represent the local maxima in the respective curves of $\mathrm{D}{\mathrm{F}}_{\max }$.

Figure 7

(a) Current-pressure (I-P) curves are plotted for a cylindrical nanopore ($\theta ={90}^{\circ }$) with $\sigma =-0.03\mathrm{C}/{\mathrm{m}}^2$, $\varepsilon =12$, $R_{\mathrm{t}}=8$ nm, $h=55$ nm, $c_0=100$ mM, and $\delta =0$. Here, the (black) dashed, (red) square-dotted, (green) solid, (orange) long-dashed, and (blue) dashed-dotted lines correspond to the $\mathrm{\Delta }V$ values of 1, 0.5, 0, −0.5, and −1 V, respectively. The subsequent consideration of $\delta \ne 0$ yields nonmonotonic I-P plots in (b) and the associated streaming current vs pressure ($I_{\mathrm{str}}\text{−}P$) plots in (c)–(e). The horizontal axes of (c) and (d) have identical limits. (f) Current-voltage curves for a conical nanopore without a membrane domain are plotted at varying pressures ($\mathrm{\Delta }P$) with $\sigma =-0.026\mathrm{C}/{\mathrm{m}}^2$, $R_{\mathrm{t}}=185$ nm, $h=20\mu \mathrm{m}$, $\theta ={80}^{\circ }$, and $c_0=10$ mM. (g) $I_{\mathrm{str}}\text{−}P$ plots for a graphene nanopore with $\sigma =0$, $\varepsilon =3$, $R_{\mathrm{t}}=1.16$ nm, $h=0.34$ nm, $c_0=1$ M, $\mathrm{\Delta }V=100$ mV, $D\left({\mathrm{K}}^{+}\right)=1.45\times {10}^{-9}{\mathrm{m}}^2/\mathrm{s}$, $D$(${\mathrm{Cl}}^{-}$)$=1.51\times {10}^{-9}{\mathrm{m}}^2/\mathrm{s}$, and slip velocities on the membrane surfaces (detailed in Sec. IV of the Supplemental Material [78]). The left and right insets in (g) demonstrate the impact of negative and positive $\mathrm{\Delta }P$ on the net ionic concentrations ($c_{+}-c_{-}$) near the pore opening, respectively.