Effect of geometry on concentration polarization in realistic heterogeneous permselective systems

Ion concentration polarization causes a nearly pore-length-independent conductance of nanopores

There has been a great amount of interest in nanopores as the basis for sensors and templates for preparation of biomimetic channels as well as model systems to understand transport properties at the nanoscale. The presence of surface charges on the pore walls has been shown to induce ion selectivity as well as enhance ionic conductance compared to uncharged pores. Here, using three-dimensional continuum modeling, we examine the role of the length of charged nanopores as well as applied voltage for controlling ion selectivity and ionic conductance of single nanopores and small nanopore arrays. First, we present conditions where the ion current and ion selectivity of nanopores with homogeneous surface charges remain unchanged, even if the pore length decreases by a factor of 6. This length-independent conductance is explained through the effect of ion concentration polarization (ICP), which modifies local ionic concentrations, not only at the pore entrances but also in the pore in a voltage-dependent manner. We describe how voltage controls the ion selectivity of nanopores with different lengths and present the conditions when charged nanopores conduct less current than uncharged pores of the same geometrical characteristics. The manuscript provides different measures of the extent of the depletion zone induced by ICP in single pores and nanopore arrays, including systems with ionic diodes. The modeling shown here will help design selective nanopores for a variety of applications where single nanopores and nanopore arrays are used.

Advances in Two-Dimensional Ion-Selective Membranes: Bridging Nanoscale Insights to Industrial-Scale Salinity Gradient Energy Harvesting

Salinity gradient energy, often referred to as the Gibbs free energy difference between saltwater and freshwater, is recognized as "blue energy" due to its inherent cleanliness, renewability, and continuous availability. Reverse electrodialysis (RED), relying on ion-selective membranes, stands as one of the most prevalent and promising methods for harnessing salinity gradient energy to generate electricity. Nevertheless, conventional RED membranes face challenges such as insufficient ion selectivity and transport rates and the difficulty of achieving the minimum commercial energy density threshold of 5 W/m2. In contrast, two-dimensional nanostructured materials, featuring nanoscale channels and abundant functional groups, offer a breakthrough by facilitating rapid ion transport and heightened selectivity. This comprehensive review delves into the mechanisms of osmotic power generation within a single nanopore and nanochannel, exploring optimal nanopore dimensions and nanochannel lengths. We subsequently examine the current landscape of power generation using two-dimensional nanostructured materials in laboratory-scale settings across various test areas. Furthermore, we address the notable decline in power density observed as test areas expand and propose essential criteria for the industrialization of two-dimensional ion-selective membranes. The review concludes with a forward-looking perspective, outlining future research directions, including scalable membrane fabrication, enhanced environmental adaptability, and integration into multiple industries. This review aims to bridge the gap between previous laboratory-scale investigations of two-dimensional ion-selective membranes in salinity gradient energy conversion and their potential large-scale industrial applications.

Electroneutrality breakdown in nanopore arrays

The electrostatic screening of charge in one-dimensional confinement leads to long-range breakdown in electroneutrality within a nanopore. Through a series of continuum simulations, we demonstrate the principles of electroneutrality breakdown for electrolytes in one-dimensional confinement. We show how interacting pores in a membrane can counteract the phenomenon of electroneutrality breakdown, eventually returning to electroneutrality. Emphasis is placed on applying simplifying formulas to reduce the multidimensional partial differential equations into a single ordinary differential equation for the electrostatic potential. Dielectric mismatch between the electrolyte and membrane, pore aspect ratio, and confinement dimensionality are studied independently, outlining the relevance of electroneutrality breakdown in nanoporous membranes for selective ion transport and separations.

Conditions for electroneutrality breakdown in nanopores

It has recently been suggested that a breakdown of electroneutrality occurs in highly confined nanopores that are encompassed by a dielectric material. This work elucidates the conditions for this breakdown. We show that the breakdown within the pore results from the response of the electric field within the dielectric. Namely, we show that this response is highly sensitive to the boundary condition at the dielectric edge. The standard Neumann boundary condition of no-flux predicts that the breakdown does not occur. However, a Dirichlet boundary condition for a zero-potential predicts a breakdown. Within this latter scenario, the breakdown exhibits a dependence on the thickness of the dielectric material. Specifically, infinite thickness dielectrics do not exhibit a breakdown, while dielectrics of finite thickness do exhibit a breakdown. Numerical simulations confirm theoretical predictions. The breakdown outcomes are discussed with regard to single pore systems and multiple pore systems.

Inertial focusing and zeta potential measurements of single-nanoparticles using octet-nanochannels

Capture-to-translocation dynamics control is an important issue for single-particle and -molecule analyses by resistive pulse waveforms. Here, we report on regulated motions for accurate zeta-potential assessments of single nanoscale objects passing through an octet-nanochannel. We observed ionic spike signals consisting of eight consecutive sub-pulses signifying the ion blockage at the eight sensing zones in series upon electrophoretic translocation of individual nanoparticles. We find an exponential decrease to saturation of the channel-to-channel translocation duration as a nanobead moves forward, reflecting the more restricted radial motion degrees of freedom via inertial effects at the downstream side of the octet channel. This finding enabled a protocol for single-nanoparticle zeta potential estimation impervious to the uncertainty stemming from the stochastic nature of the translocation dynamics. The multi-channel approach presented in this study may be used as a useful tool for analyzing particles and molecules of variable sizes.

Bipolar Nanochannels: A Systematic Approach to Asymmetric Problems

Nanofluidic diodes are capable of rectifying the electrical current by several orders of magnitude. In the current state of affairs, determining the rectification factor is not possible as it depends on many system parameters. In this work, we systematically scan the effects of geometry and excess counterion concentrations (i.e., surface charge effects). We show that the current-voltage response varies between the two extreme behaviors of unipolar and bipolar responses. The exact behavior depends on the geometry and surface charge properties of the system. Here, we have gone beyond the typical setup that only considers the dynamics within the nanochannel itself and we have included the effects of the adjoining microchannels. Systems that include both nanochannels and microchannels exhibit the classical signatures of concentration polarization, such as ionic depletion and enrichment. Here, where we have scanned a wide range of parameters, we show that bipolar and semi-bipolar systems exhibit a wider range of phenomena that are intrinsically more complicated. Our system characterization is for both, the much more investigated case of steady state and the less investigated, but equally interesting, time-transient case. For example, it is common to characterize the system by its steady-state result (current-voltage response, rectification factor, and transport number). Here, we demonstrate that the time-transient behavior of the fluxes can also be used to characterize the system, and that the time-dependent rectification factors and transport numbers are meaningful. The systematic approach taken in this work, and the results presented herein, can be used to further elucidate the complicated behavior of the current-voltage response of nanofluidic diodes and to rationalize experimental results. The insights of this work can be used to enhance and improve the design of all nanofluidic diodes.

Ion transport in nanopores with highly overlapping electric double layers

Investigation of ion transport through nanopores with highly overlapping electric double layers is extremely challenging. This can be attributed to the non-linear Poisson-Boltzmann equation that governs the behavior of the electrical potential distribution as well as other characteristics of ion transport. In this work, we leverage the approach of Schnitzer and Yariv [Phys. Rev. E 87, 054301 (2013)] to reduce the complexity of the governing equation. An asymptotic solution is derived, which shows remarkable correspondence to simulations of the non-approximated equations. This new solution is leveraged to address a number of highly debated issues. We derive the equivalent of the Gouy-Chapman equation for systems with highly overlapping electric double layers. This new relationship between the surface charge density and the surface potential is then utilized to determine the power-law scaling of nanopore conductances as a function of the bulk concentrations. We derive the coefficients of transport for the case of overlapping electric double layers and compare it to the renowned uniform potential model. We show that the uniform potential model is only an approximation for the exact solution for small surface charges. The findings of this work can be leveraged to uncover additional hidden attributes of ion transport through nanopores.

Controllable pH Manipulations in Micro/Nanofluidic Device Using Nanoscale Electrokinetics

Recently introduced nanoscale electrokinetic phenomenon called ion concentration polarization (ICP) has been suffered from serious pH changes to the sample fluid. A number of studies have focused on the origin of pH changes and strategies for regulating it. Instead of avoiding pH changes, in this work, we tried to demonstrate new ways to utilize this inevitable pH change. First, one can obtain a well-defined pH gradient in proton-received microchannel by applying a fixed electric current through a proton exchange membrane. Furthermore, one can tune the pH gradient on demand by adjusting the proton mass transportation (i.e., adjusting electric current). Secondly, we demonstrated that the occurrence of ICP can be examined by sensing a surrounding pH of electrolyte solution. When pH > threshold pH, patterned pH-responsive hydrogel inside a straight microchannel acted as a nanojunction to block the microchannel, while it did as a microjunction when pH < threshold pH. In case of forming a nanojunction, electrical current significantly dropped compared to the case of a microjunction. The strategies that presented in this work would be a basis for useful engineering applications such as a localized pH stimulation to biomolecules using tunable pH gradient generation and portable pH sensor with pH-sensitive hydrogel.

Dynamics of driftless preconcentration using ion concentration polarization leveraged by convection and diffusion

Over the past several decades, separation and preconcentration methods of (bio)molecules have been actively developed for various biomedical and chemical processes such as disease diagnostics, point of care test and environmental monitoring. Among the great developments of the electrokinetic method in a micro/nanofluidic platform is the ion concentration polarization (ICP) phenomenon, in which a target molecule is accumulated near a permselective nanoporous membrane under an applied electric field. ICP method has been actively studied due to its easy implementation and high preconcentration/separation efficiency. However, the dynamic behavior of preconcentrated analytes has not yet been fully studied, especially driftless migration, where the applied electric field is orthogonal to the direction of the drift migration. Here, we demonstrate anomalous shapes of preconcentrated analytes (either plug or dumbbell shape) and the morphologies were analytically modeled by the leverage of convection and diffusion migration. This model was experimentally verified with various lengths of DNA and the limiting cases (convection-free environment in paper-based microfluidic device and extremely low diffusivity of red blood cells) were also shown to confirm the model. Thus, this study not only provides an insight into the fundamental electrokinetic dynamics of molecules in an ICP platform but also plays a guiding role for the design of a nanofluidic preconcentrator for a lab on a chip application.

Charge polarization, local electroneutrality breakdown and eddy formation due to electroosmosis in varying-section channels

We characterize the dynamics of an electrolyte embedded in a varying-section channel under the action of a constant external electrostatic field. By means of molecular dynamics simulations we determine the stationary density, charge and velocity profiles of the electrolyte. Our results show that when the Debye length is comparable to the width of the channel bottlenecks a concentration polarization along with two eddies sets inside the channel. Interestingly, upon increasing the external field, local electroneutrality breaks down and charge polarization sets leading to the onset of net dipolar field. This novel scenario, that cannot be captured by the standard approaches based on local electroneutrality, opens the route for the realization of novel micro and nano-fluidic devices.

A concentration-independent micro/nanofluidic active diode using an asymmetric ion concentration polarization layer

Over the past decade, nanofluidic diodes that rectify ionic currents (i.e. greater current in one direction than in the opposite direction) have drawn significant attention in biomolecular sensing, switching and energy harvesting devices. To obtain current rectification, conventional nanofluidic diodes have utilized complex nanoscale asymmetry such as nanochannel geometry, surface charge density, and reservoir concentration. Avoiding the use of sophisticated nano-asymmetry, micro/nanofluidic diodes using microscale asymmetry have been recently introduced; however, their diodic performance is still impeded by (i) low (even absent) rectification effects at physiological concentrations over 100 mM and strong dependency on the bulk concentration, and (ii) the fact that they possess only passive predefined rectification factors. Here, we demonstrated a new class of micro/nanofluidic diode with an ideal perm-selective nanoporous membrane based on ion concentration polarization (ICP) phenomenon. Thin side-microchannels installed near a nanojunction served as mitigators of the amplified electrokinetic flows generated by ICP and induced convective salt transfer to the nanoporous membrane, leading to actively controlled micro-scale asymmetry. Using this device, current rectifications were successfully demonstrated in a wide range of electrolytic concentrations (10^-5 M to 3 M) as a function of the fluidic resistance of the side-microchannels. Noteworthily, it was confirmed that the rectification factors were independent from the bulk concentration due to the ideal perm-selectivity. Moreover, the rectification of the presenting diode was actively controlled by adjusting the external convective flows, while that of the previous diode was passively determined by invariant nanoscale asymmetry.

Transport-Induced Inversion of Screening Ionic Charges in Nanochannels

This work reveals a counterintuitive but basic process of ionic screening in nanofluidic channels. Steady-state numerical simulations and mathematical analysis show that, under significant longitudinal ionic transport, the screening ionic charges can be locally inverted in the channels: their charge sign becomes the same as that of the channel surface charges. The process is identified to originate from the coupling of ionic electro-diffusion transport and junction two-dimensional electrostatics. This finding may expand our understanding of ionic screening and electrical double layers in nanochannels. Furthermore, the charge inversion process results in a body-force torque on channel fluids, which is a possible mechanism for vortex generation in the channels and their nonlinear current-voltage characteristics.

Effect of field-focusing and ion selectivity on the extended space charge developed at the microchannel-nanochannel interface

We present results demonstrating the effect of varying microchannel depth and bulk conductivity on the space charge-mediated transition between classical, diffusion-limited current and over-limiting current in microchannel-nanochannel devices. The extended space charge layer develops at the depleted microchannel-nanochannel entrance when the limiting current is exceeded and is correlated with a distinctive maximum in the dc resistance. This maximum is shown to be affected by the microchannel depth, via field-focusing, and solution conductivity. In particular, we observe that upon their increase, the maximum becomes flatter and shifts to higher voltages.

Induced-charge electrokinetics, bipolar current, and concentration polarization in a microchannel-Nafion-membrane system

The presence of a floating electrode array located within the depletion layer formed due to concentration polarization across a microchannel-membrane interface device may produce not only induced-charge electro-osmosis (ICEO) but also bipolar current resulting from the induced Faradaic reaction. It has been shown that there exists an optimal thickness of a thin dielectric coating that is sufficient to suppress bipolar currents but still enables ICEO vortices that stir the depletion layer, thereby affecting the system's current-voltage response. In addition, the use of alternating-current electro-osmosis by activating electrodes results in further enhancement of the fluid stirring and opens new routes for on-demand spatiotemporal control of the depletion layer length.

Interplay between Nanochannel and Microchannel Resistances

Current nanochannel system paradigm commonly neglects the role of the interfacing microchannels and assumes that the ohmic electrical response of a microchannel-nanochannel system is solely determined by the geometric properties of the nanochannel. In this work, we demonstrate that the overall response is determined by the interplay between the nanochannel resistance and various microchannel attributed resistances. Our experiments confirm a recent theoretical prediction that in contrast to what was previously assumed at very low concentrations the role of the interfacing microchannels on the overall resistance becomes increasingly important. We argue that the current nanochannel-dominated conductance paradigm can be replaced with a more correct and intuitive microchannel-nanochannel-resistance-model-based paradigm.

Asymmetry-induced electric current rectification in permselective systems

For a symmetric ion permselective system, in terms of geometry and bulk concentrations, the system response is also symmetric under opposite electric field polarity. In this work we derive an analytical solution for the concentration distribution, electric potential, and current-voltage response for a four-layered system comprised of two microchambers connected by two permselective regions of varying properties. It is shown that any additional asymmetry in the system, in terms of the geometry, bulk concentration, or surface charge property of the permselective regions, results in current rectification. Our work is divided into two parts: when both permselective regions have the same surface charge sign and the case of opposite signs. For the same sign case we are able to show that the system behaves as a dialytic battery while accounting for field-focusing effects. For the case of opposite signs (i.e., bipolar membrane), our system exhibits the behavior of a bipolar diode where the magnitude of the rectification can be of order 10^{2}-10^{3}.

Time-dependent ion transport in heterogeneous permselective systems

The current study extends previous analytical and numerical solutions of chronopotentiometric response of one-dimensional systems consisting of three layers to the more realistic two-dimensional (2D) heterogeneous ion-permselective medium. An analytical solution for the transient concentration-polarization problem, under the local electroneutrality approximation and assumption of ideal permselectivity, was obtained using the Laplace transform and separation of variables technique. Then the 2D electric potential was obtained numerically and was compared to the full Poisson-Nernst-Planck solution. It was then shown that the resultant voltage drop across the system varies between the initial Ohmic response and that of the steady state accounting for concentration polarization. Also, the field-focusing effect in a 2D system is shown to result in a faster depletion of ions at the permselective interface.

Selective preconcentration and online collection of charged molecules using ion concentration polarization

Online collection of selectivity preconcentrated analytes was demonstrated utilizing ion concentration polarization phenomena and pneumatic valve system.

Bridging the gap between an isolated nanochannel and a communicating multipore heterogeneous membrane

To bridge the gap between single and isolated pore systems to multipore systems, such as membranes and electrodes, we studied an array of nanochannels with varying interchannel spacing that controlled the degree of channel communication. Instead of treating them as individual channels connected in parallel or an assembly like a homogeneous membrane, this study resolves the pore-pore interaction. We found that increased channel isolation leads to current intensification, whereas at high voltages electroconvective effects control the degree of communication via suppression of the diffusion layer growth.