Electrokinetic Effects on Flow and Ion Transport in Charge-Patterned Corrugated Nanochannels

TL;DR

This work addresses how charge-patterned, corrugated nanochannels control flow and ion transport under pressure-driven and electrokinetically influenced conditions. By solving the full nonlinear Poisson–Nernst–Planck–Stokes equations (PNPS) for periodic geometries and surface charge distributions, the authors reveal surface-gradient induced electroosmosis (SGIEO) as a mechanism that generates finite flow in the absence of macroscopic driving forces. They identify three flow regimes as the applied pressure gradient grows, showing transitions from electrokinetically inhibited to pressure-dominated transport and demonstrating diode-like ionic currents and high selectivity near regime transitions. The findings offer design principles for tunable flow control and selective ion transport in nanochannels, with implications for desalination, energy harvesting, and microfluidic pumping strategies, and point to future work on AC/DC control and dynamic boundary effects. All mathematical notation is presented with $...$ delimiters to preserve clarity of the analytical framework.

Abstract

This study explores how the distribution of surface charge along corrugated nanochannels affects flow rates and influences ionic currents and charge selectivity in a pressure gradient-driven flow. We numerically solve the coupled Poisson-Nernst-Planck-Stokes (PNPS) equations for periodic aperture profiles and explore how changes in the Debye screening length and the degree of symmetry between surface charge and geometry, and the magnitude of an applied pressure gradient affect the velocity profile. We identify three regimes of flow: I. At low (no) pressure gradients, the inhomogeneous distribution of surface charge generates a nonlinear torque that drives recirculating flow. Placing surface charge asymmetrically with respect to the geometry produces a net axial flow. II. At moderate pressure gradients, the flow rate is proportional to the mechanical driving force, though is significantly diminished relative to channels absent of surface charge inhomogeneity. Throughput is inhibited by the electrostatic force that opposes the displacement of ions from the diffuse part of the electric double layer. III. At high pressure gradients, we demonstrate a transition between electrostatically and mechanically controlled flow regimes, where -- under appropriate choice of parameters -- a marginal increase in the applied pressure gradient triggers an abrupt, orders-of-magnitude increase in the mean velocity. By incorporating the computed velocity and electric fields from the PNPS equations into a random walk particle tracking algorithm, we provide a detailed quantitative characterization of the transport dynamics of the ions, demonstrating the ability to selectively control the flux of charge and moderate the rate of ion dispersion.

Figures (11)

• Figure 1: (a) Concept diagram of electrochemical flow through a wavy channel with spatially varying surface charge density. The orange spheres represent Brownian tracer particles advected by the flow. Numerical discretization of the field variables in (b) the physical domain and (c) the transformed domain; grid point resolution is reduced for clarity. In a staggered arrangement, the velocity, electrostatic potential, and ion concentrations are evaluated at the orange points, and the pressure is evaluated at the purple points. Using the problem's symmetry, the governing equations are evaluated numerically in the domain of a half-pore.
• Figure 2: (a) Vorticity generating drift force along a sinusoidal patch of charge (a half-sine wave) at $\tilde{y}=1/2$ in a flat channel; the inset shows the power-law scaling of the amplitude of the source as a function of the Debye screening length. (b,c) Electrostatic potential profiles across a single patch of charge in the volume of a flat channel for low salt concentration ((b): $c_0=0.5$ M and $2\tilde{l}_\mathrm{D} = 0.16$) and high salt concentration ((c): $c_0=0.005$ M and $2\tilde{l}_\mathrm{D}=1.64$). The top surface of the channel is located at $\tilde{y}=1/2$ and the centerline axis of the channel at $\tilde{y}=0$. Solid contours represent the steady-state profiles measured by solving the system of PNPS equations, while dashed contours plot the equilibrium profiles of the Poisson-Boltzmann equation in the DH approximation in equation (\ref{['eq:DH_potential']}). Background colormaps display the vorticity in the steady-state flow field.
• Figure 3: (a-e) Velocity fields induced by SGIEO in the absence of external pressure gradients (with $\nabla p_0 =0\,\mathrm{Pa}$) in the top half of a pore across one wavelength of the geometric undulations. Profiles correspond to $\delta \tilde{W}=0.5$ and $c_0=0.05\,\mathrm{M}$; the remaining parameters for the simulation are provided in table \ref{['tab:parameter_values']}. The grayscale colormap displays the magnitude of the velocity profiles with streamlines indicating the direction of flow. The colored line along the top boundary indicates the prescribed surface charge density -- purple indicates positive charge and orange indicates negative charge -- which is shifted along the channel axis moving from (a) to (e). (f-j) Profiles of the normalised Okubo-Weiss parameter, $\tilde{\Theta}$, with streamlines indicating the alignment of the electric field, $\tilde{\boldsymbol{E}}$.
• Figure 4: (a-e) Velocity fields for SGIEO flow with $k=1$, $\delta \tilde{W}=0.5$, $\nabla p_0=0\,\mathrm{Pa}$, and $\varphi=\pi/4$. The Debye screening length is adjusted with values -- moving from top to bottom -- of $2\tilde{l}_\mathrm{D}=0.16,\,0.52$ and $1.64$ (these correspond to salt $c_0=0.5\,\mathrm{M},\, 0.05\,\mathrm{M},$ and $0.005\,\mathrm{M}$, respectively). (f-j) Distributions of the dimensionless electrokinetic torque, $\tilde{S}_\omega$, plotted beneath the electric field lines. The colored line along the top boundary indicates the prescribed surface charge density where purple indicates positive charge and orange indicates negative charge.
• Figure 5: Plots similar to those displayed in figure \ref{['fig:velocity_profiles_Debye']} for $k=0.5$, $\varphi=3\pi/4$, and Debye lengths of $2\tilde{l}_\mathrm{D}=0.16,\,0.52$ and $1.64$ -- corresponding to $c_0=0.5\,\mathrm{M}, 0.1\,\mathrm{M},$ and $0.05\,\mathrm{M}$, respectively; the distribution of positive surface charge is plotted in purple along the top channel boundary in (a,b,c).