Crossover of the Power-Law Exponent for Carbon Nanotube Conductivity as a Function of Salinity

Scaling Behavior and Conductance Mechanisms of Ion Transport in Atomically Thin Graphene Nano/Subnanopores

Ion transport through atomically thin nano/subnanopores, such as those in monolayer graphene, presents challenges to traditional ion conduction models, primarily due to extreme confinement effects and hydration interactions. Under these conditions, existing models fail to account for conductance behaviors at the nano- and subnanometer scales. In this study, we perform a combined experimental and theoretical investigation of ion transport in monolayer graphene nano/subnanopores across varying salt concentrations. We introduce a conductance model that accurately predicts the observed scaling behavior by addressing the interaction between counterions and the edges of atomically thin pores, where counterion movement is constrained by the pore's structure. This model also quantifies the hydration energy barrier, highlighting the impact of the hydration shell structures on ion transport efficiency. Our findings reveal that hydrated potassium ions traverse these pores with higher efficiency than previously estimated, offering new insights into ion transport mechanisms under atomic-scale confinement.

Emerging Advances around Nanofluidic Transport and Mass Separation under Confinement in Atomically Thin Nanoporous Graphene

Membrane separation stands as an environmentally friendly, high permeance and selectivity, low energy demand process that deserves scientific investigation and industrialization. To address intensive demand, seeking appropriate membrane materials to surpass trade-off between permeability and selectivity and improve stability is on the schedule. 2D materials offer transformational opportunities and a revolutionary platform for researching membrane separation process. Especially, the atomically thin graphene with controllable porosity and structure, as well as unique properties, is widely considered as a candidate for membrane materials aiming to provide extreme stability, exponentially large selectivity combined with high permeability. Currently, it has shown promising opportunities to develop separation membranes to tackle bottlenecks of traditional membranes, and it has been of great interest for tremendously versatile applications such as separation, energy harvesting, and sensing. In this review, starting from transport mechanisms of separation, the material selection bank is narrowed down to nanoporous graphene. The study presents an enlightening overview of very recent developments in the preparation of atomically thin nanoporous graphene and correlates surface properties of such 2D nanoporous materials to their performance in critical separation applications. Finally, challenges related to modulation and manufacturing as well as potential avenues for performance improvements are also pointed out.

Zeta Potentials of Cotton Membranes in Acetonitrile Solutions

Solid surfaces in contact with nonaqueous solvents play a key role in electrochemistry, analytical chemistry, and industrial chemistry. In this work, the zeta potentials of cotton membranes in acetonitrile solutions were determined by streaming potential and bulk conductivity measurements. By applying the Gouy-Chapman theory and the Langmuir adsorption isotherm of ions to the experimental data, the mechanism of the electrification at the cotton/acetonitrile interface is revealed for the first time to be solely due to ion adsorption on the surface, rather than proton dissociation at the interface. Different salts were found to produce opposite signs of the zeta potentials. This behavior can be attributed to ion solvation effects and the strong ordering of acetonitrile molecules at the interface. Furthermore, a trend of the electroviscous effect was observed, in agreement with the standard electrokinetic theory. These findings demonstrate that electrokinetics in acetonitrile, a polar aprotic solvent, can be treated in the same manner as in water.

Electrophoresis of ions and electrolyte conductivity: From bulk to nanochannels

When electrolyte solutions are confined in micro- and nanochannels their conductivity is significantly different from those in a bulk phase. Here we revisit the theory of this phenomenon by focusing attention on the reduction in the ion mobility with the concentration of salt and a consequent impact to the conductivity of a monovalent solution, from bulk to confined in a narrow slit. We first give a systematic treatment of electrophoresis of ions and obtain equations for their zeta potentials and mobilities. The latter are then used to obtain a simple expression for a bulk conductivity, which is valid in a concentration range up to a few molars and more accurate than prior analytic theories. By extending the formalism to the electrolyte solution in the charged channel the equations describing the conductivity in different modes are presented. They can be regarded as a generalization of prior work on the channel conductivity to a more realistic case of a nonzero reduction of the electrophoretic mobility of ions with salt concentration. Our analysis provides a framework for interpreting measurements on the conductivity of electrolyte solutions in the bulk and in narrow channels.

Fluids and Electrolytes under Confinement in Single-Digit Nanopores

Confined fluids and electrolyte solutions in nanopores exhibit rich and surprising physics and chemistry that impact the mass transport and energy efficiency in many important natural systems and industrial applications. Existing theories often fail to predict the exotic effects observed in the narrowest of such pores, called single-digit nanopores (SDNs), which have diameters or conduit widths of less than 10 nm, and have only recently become accessible for experimental measurements. What SDNs reveal has been surprising, including a rapidly increasing number of examples such as extraordinarily fast water transport, distorted fluid-phase boundaries, strong ion-correlation and quantum effects, and dielectric anomalies that are not observed in larger pores. Exploiting these effects presents myriad opportunities in both basic and applied research that stand to impact a host of new technologies at the water-energy nexus, from new membranes for precise separations and water purification to new gas permeable materials for water electrolyzers and energy-storage devices. SDNs also present unique opportunities to achieve ultrasensitive and selective chemical sensing at the single-ion and single-molecule limit. In this review article, we summarize the progress on nanofluidics of SDNs, with a focus on the confinement effects that arise in these extremely narrow nanopores. The recent development of precision model systems, transformative experimental tools, and multiscale theories that have played enabling roles in advancing this frontier are reviewed. We also identify new knowledge gaps in our understanding of nanofluidic transport and provide an outlook for the future challenges and opportunities at this rapidly advancing frontier.

Electrical Conductance of Charged Nanopores

A nanopore's response to an electrical potential drop is characterized by its electrical conductance, . For the last two decades, it has been thought that at low electrolyte concentrations, , the conductance is concentration-independent such that . It has been recently demonstrated that surface charge regulation changes the dependency to , whereby the slope typically takes the values α = 1/3 or 1/2. However, experiments have observed slopes of 2/3 and 1 suggesting that additional mechanisms, such as convection and slip-lengths, appear. Here, we elucidate the interplay between three mechanisms: surface charge regulation, convection, and slip lengths. We show that the inclusion of convection does not change the slope, and when the effects of hydrodynamic slip are included, the slope is doubled. We show that when all effects are accounted for, α can take any value between 0 and 1 where the exact value of the slope depends on the material properties. This result is of utmost importance in designing any electro-kinetically driven nanofluidic system characterized by its conductance.

Putting together the puzzle of ion transfer in single-digit carbon nanotubes: mean-field meets ab initio

Nature employs channel proteins to selectively pass water across cell membranes, which inspires the search for bio-mimetic analogues. Carbon nanotube porins (CNTPs) are intriguing mimics of water channels, yet ion transport in CNTPs still poses questions. As an alternative to continuum models, here we present a molecular mean-field model that transparently describes ion coupling, yet unlike continuum models, computes ab initio all required thermodynamic quantities for the KCl salt and H⁺ and OH^- ions present in water. Starting from water transfer, the model considers the transfer of free ions, along with ion-pair formation as a proxy of non-mean-field ion-ion interactions. High affinity to hydroxide, suggested by experiments, making it a dominant charge carrier in CNTPs, is revealed as an exceptionally favorable transfer of KOH pairs. Nevertheless, free ions, coexisting with less mobile ion-pairs, apparently control ion transport. The model well explains the observed effects of salt concentration and pH on conductivity, transport numbers, anion permeation and its activation energies, and current rectification. The proposed approach is extendable to other sub-nanochannels and helps design novel osmotic materials and devices.

Enhanced nanofluidic transport in activated carbon nanoconduits

Carbon has emerged as a unique material in nanofluidics, with reports of fast water transport, molecular ion separation and efficient osmotic energy conversion. Many of these phenomena still await proper rationalization due to the lack of fundamental understanding of nanoscale ionic transport, which can only be achieved in controlled environments. Here we develop the fabrication of 'activated' two-dimensional carbon nanochannels. Compared with nanoconduits with 'pristine' graphite walls, this enables the investigation of nanoscale ionic transport in great detail. We show that activated carbon nanochannels outperform pristine channels by orders of magnitude in terms of surface electrification, ionic conductance, streaming current and (epi-)osmotic currents. A detailed theoretical framework enables us to attribute the enhanced ionic transport across activated carbon nanochannels to an optimal combination of high surface charge and low friction. Furthermore, this demonstrates the unique potential of activated carbon for energy harvesting from salinity gradients with single-pore power density across activated carbon nanochannels, reaching hundreds of kilowatts per square metre, surpassing alternative nanomaterials.

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.

Enhanced transport of ions by tuning surface properties of the nanochannel

Motivated by recent observations of anomalously large deviations of the conductivity currents in confined systems from the bulk behavior, we revisit the theory of ion transport in parallel-plate channels and also discuss how the wettability of a solid and the mobility of adsorbed surface charges impact the transport of ions. It is shown that depending on the ratio of the electrostatic disjoining pressure to the excess osmotic pressure at the walls two different regimes occur. In the thick channel regime this ratio is small and the channel effectively behaves as thick, even when the diffuse layers strongly overlap. The latter is possible for highly charged channels only. In the thin channel regime the disjoining pressure is comparable to the excess osmotic pressure at the wall, which implies relatively weakly charged walls. We derive simple expressions for the mean conductivity of the channel in these two regimes, highlighting the role of electrostatic and electrohydrodynamic boundary conditions. Our theory provides a simple explanation of the high conductivity observed experimentally in hydrophilic channels, and allows one to obtain rigorous bounds on its attainable value and scaling with salt concentration. Our results also show that further dramatic amplification of conductivity is possible if hydrophobic slip is involved, but only in the thick channel regime provided the walls are sufficiently highly charged and most of the adsorbed charges are immobile. However, for weakly charged surfaces the massive conductivity amplification due to hydrodynamic slip is impossible in both regimes. Interestingly, in this case the moderate slip-driven contribution to conductivity can monotonously decrease with the fraction of immobile adsorbed charges. These results provide a framework for tuning the conductivity of nanochannels by adjusting their surface properties and bulk electrolyte concentrations.

Electrification of water interface

The surface charge of a water interface determines many fundamental processes in physical chemistry and interface science, and it has been intensively studied for over a hundred years. We summarize experimental methods to characterize the surface charge densities developed so far: electrokinetics, double-layer force measurements, potentiometric titration, surface-sensitive nonlinear spectroscopy, and surface-sensitive mass spectrometry. Then, we elucidate physical ion adsorption and chemical electrification as examples of electrification mechanisms. In the end, novel effects on surface electrification are discussed in detail. We believe that this clear overview of state of the art in a charged water interface will surely help the fundamental progress of physics and chemistry at interfaces in the future.

From nanotubes to nanoholes: Scaling of selectivity in uniformly charged nanopores through the Dukhin number for 1:1 electrolytes

Scaling of the behavior of a nanodevice means that the device function (selectivity) is a unique smooth and monotonic function of a scaling parameter that is an appropriate combination of the system's parameters. For the uniformly charged cylindrical nanopore studied here, these parameters are the electrolyte concentration, c, voltage, U, the radius and the length of the nanopore, R and H, and the surface charge density on the nanopore's surface, σ. Due to the non-linear dependence of selectivities on these parameters, scaling can only be applied in certain limits. We show that the Dukhin number, Du=|σ|/eRc∼|σ|λ_D ²/eR (λ_D is the Debye length), is an appropriate scaling parameter in the nanotube limit (H → ∞). Decreasing the length of the nanopore, namely, approaching the nanohole limit (H → 0), an alternative scaling parameter has been obtained, which contains the pore length and is called the modified Dukhin number: mDu ∼ Du H/λ_D ∼ |σ|λ_DH/eR. We found that the reason for non-linearity is that the double layers accumulating at the pore wall in the radial dimension correlate with the double layers accumulating at the entrances of the pore near the membrane on the two sides. Our modeling study using the Local Equilibrium Monte Carlo method and the Poisson-Nernst-Planck theory provides concentration, flux, and selectivity profiles that show whether the surface or the volume conduction dominates in a given region of the nanopore for a given combination of the variables. We propose that the inflection point of the scaling curve may be used to characterize the transition point between the surface and volume conductions.

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.

Accurate modeling of a biological nanopore with an extended continuum framework

Despite the broad success of biological nanopores as powerful instruments for the analysis of proteins and nucleic acids at the single-molecule level, a fast simulation methodology to accurately model their nanofluidic properties is currently unavailable. This limits the rational engineering of nanopore traits and makes the unambiguous interpretation of experimental results challenging. Here, we present a continuum approach that can faithfully reproduce the experimentally measured ionic conductance of the biological nanopore Cytolysin A (ClyA) over a wide range of ionic strengths and bias potentials. Our model consists of the extended Poisson-Nernst-Planck and Navier-Stokes (ePNP-NS) equations and a computationally efficient 2D-axisymmetric representation for the geometry and charge distribution of the nanopore. Importantly, the ePNP-NS equations achieve this accuracy by self-consistently considering the finite size of the ions and the influence of both the ionic strength and the nanoscopic scale of the pore on the local properties of the electrolyte. These comprise the mobility and diffusivity of the ions, and the density, viscosity and relative permittivity of the solvent. Crucially, by applying our methodology to ClyA, a biological nanopore used for single-molecule enzymology studies, we could directly quantify several nanofluidic characteristics difficult to determine experimentally. These include the ion selectivity, the ion concentration distributions, the electrostatic potential landscape, the magnitude of the electro-osmotic flow field, and the internal pressure distribution. Hence, this work provides a means to obtain fundamental new insights into the nanofluidic properties of biological nanopores and paves the way towards their rational engineering.

Ion Transport in Electrically Imperfect Nanopores

Ionic transport through a charged nanopore at low ion concentration is governed by the surface conductance. Several experiments have reported various power-law relations between the surface conductance and ion concentration, i.e., G_surf ∝ c₀^α. However, the physical origin of the varying exponent, α, is not yet clearly understood. By performing extensive coarse-grained Molecular Dynamics simulations for various pore diameters, lengths, and surface charge densities, we observe varying power-law exponents even with a constant surface charge and show that α depends on how electrically "perfect" the nanopore is. Specifically, when the net charge of the solution in the pore is insufficient to ensure electroneutrality, the pore is electrically "imperfect" and such nanopores can exhibit varying α depending on the degree of "imperfectness". We present an ionic conductance theory for electrically "imperfect" nanopores that not only explains the various power-law relationships but also describes most of the experimental data available in the literature.

Strong Electroosmotic Coupling Dominates Ion Conductance of 1.5 nm Diameter Carbon Nanotube Porins

Extreme confinement in nanometer-sized channels can alter fluid and ion transport in significant ways, leading to significant water flow enhancement and unusual ion correlation effects. These effects are especially pronounced in carbon nanotube porins (CNTPs) that combine strong confinement in the inner lumen of carbon nanotubes with the high slip flow enhancement due to smooth hydrophobic pore walls. We have studied ion transport and ion selectivity in 1.5 nm diameter CNTPs embedded in lipid membranes using a single nanopore measurement setup. Our data show that CNTPs are weakly cation selective at pH 7.5 and become nonselective at pH 3.0. Ion conductance of CNTPs exhibits an unusual 2/3 power law scaling with the ion concentration at both neutral and acidic pH values. Coupled Navier-Stokes and Poisson-Nernst-Planck simulations and atomistic molecular dynamics simulations reveal that this scaling originates from strong coupling between water and ion transport in these channels. These effects could result in development of a next generation of biomimetic membranes and carbon nanotube-based electroosmotic pumps.

Interfacial transport with mobile surface charges and consequences for ionic transport in carbon nanotubes

In this paper, we explore the effect of a finite surface charge mobility on the interfacial transport: conductance, streaming currents, electro- and diffusio-osmotic flows. We first show that the surface charge mobility modifies the hydrodynamic boundary condition for the fluid, which introduces a supplementary term depending on the applied electric field. In particular, the resulting slip length is found to decrease inversely with the surface charge. We then derive expressions for the various transport mobilities, highlighting that the surface charge mobility merely moderates the amplification effect of interfacial slippage, to the noticeable exception of diffusio-osmosis and surface conductance. Our calculations, obtained within Poisson-Boltzmann framework, highlight the importance of non-linear electrostatic contributions to predict the small concentration/large charge limiting regimes for the transport mobilities. We discuss these predictions in the context of recent electrokinetic experiments with carbon nanotubes.

Role of charge regulation and flow slip in the ionic conductance of nanopores: An analytical approach

The number of precise conductance measurements in nanopores is quickly growing. To clarify the dominant mechanisms at play and facilitate the characterization of such systems for which there is still no clear consensus, we propose an analytical approach to the ionic conductance in nanopores that takes into account (i) electro-osmotic effects, (ii) flow slip at the pore surface for hydrophobic nanopores, (iii) a component of the surface charge density that is modulated by the reservoir pH and salt concentration c_{s} using a simple charge regulation model, and (iv) a fixed surface charge density that is unaffected by pH and c_{s}. Limiting cases are explored for various ranges of salt concentration and our formula is used to fit conductance experiments found in the literature for carbon nanotubes. This approach permits us to catalog the different possible transport regimes and propose an explanation for the wide variety of currently known experimental behavior for the conductance versus c_{s}.