Electro-osmotic flow through nanopores in thin and ultrathin membranes

Spiers Memorial Lecture: Towards understanding of iontronic systems: electroosmotic flow of monovalent and divalent electrolyte through charged cylindrical nanopores

In many practical applications, ions are the primary charge carrier and must move through either semipermeable membranes or through pores, which mimic ion channels in biological systems. In analogy to electronic devices, the "iontronic" ones use electric fields to induce the charge motion. However, unlike the electrons that move through a conductor, motion of ions is usually associated with simultaneous solvent flow. A study of electroosmotic flow through narrow pores is an outstanding challenge that lies at the interface of non-equilibrium statistical mechanics and fluid dynamics. In this paper, we will review recent works that use dissipative particle dynamics simulations to tackle this difficult problem. We will also present a classical density functional theory (DFT) based on the hypernetted-chain approximation (HNC), which allows us to calculate the velocity of electroosmotic flows inside nanopores containing 1 : 1 or 2 : 1 electrolyte solution. The theoretical results will be compared with simulations. In simulations, the electrostatic interactions are treated using the recently introduced pseudo-1D Ewald summation method. The zeta potentials calculated from the location of the shear plane of a pure solvent are found to agree reasonably well with the Smoluchowski equation. However, the quantitative structure of the fluid velocity profiles deviates significantly from the predictions of the Smoluchowski equation in the case of charged pores with 2 : 1 electrolyte. For low to moderate surface charge densities, the DFT allows us to accurately calculate the electrostatic potential profiles and the zeta potentials inside the nanopores. For pores with 1 : 1 electrolyte, the agreement between theory and simulation is particularly good for large ions, for which steric effects dominate over the ionic electrostatic correlations. The electroosmotic flow is found to depend very strongly on the ionic radii. In the case of pores containing 2 : 1 electrolyte, we observe a reentrant transition in which the electroosmotic flow first reverses and then returns to normal as the surface change density of the pore is increased.

Closed Formula for Transport across Constrictions

In the last decade, the Fick-Jacobs approximation has been exploited to capture transport across constrictions. Here, we review the derivation of the Fick-Jacobs equation with particular emphasis on its linear response regime. We show that, for fore-aft symmetric channels, the flux of noninteracting systems is fully captured by its linear response regime. For this case, we derive a very simple formula that captures the correct trends and can be exploited as a simple tool to design experiments or simulations. Lastly, we show that higher-order corrections in the flux may appear for nonsymmetric channels.

Ion and Water Transport in Ion-Exchange Membranes for Power Generation Systems: Guidelines for Modeling

Artificial ion-exchange and other charged membranes, such as biomembranes, are self-organizing nanomaterials built from macromolecules. The interactions of fragments of macromolecules results in phase separation and the formation of ion-conducting channels. The properties conditioned by the structure of charged membranes determine their application in separation processes (water treatment, electrolyte concentration, food industry and others), energy (reverse electrodialysis, fuel cells and others), and chlore-alkali production and others. The purpose of this review is to provide guidelines for modeling the transport of ions and water in charged membranes, as well as to describe the latest advances in this field with a focus on power generation systems. We briefly describe the main structural elements of charged membranes which determine their ion and water transport characteristics. The main governing equations and the most commonly used theories and assumptions are presented and analyzed. The known models are classified and then described based on the information about the equations and the assumptions they are based on. Most attention is paid to the models which have the greatest impact and are most frequently used in the literature. Among them, we focus on recent models developed for proton-exchange membranes used in fuel cells and for membranes applied in reverse electrodialysis.

Single-File Translocation Dynamics of SDS-Denatured, Whole Proteins through Sub-5 nm Solid-State Nanopores

The ability to routinely identify and quantify the complete proteome from single cells will greatly advance medicine and basic biology research. To meet this challenge of single-cell proteomics, single-molecule technologies are being developed and improved. Most approaches, to date, rely on the analysis of polypeptides, resulting from digested proteins, either in solution or immobilized on a surface. Nanopore biosensing is an emerging single-molecule technique that circumvents surface immobilization and is optimally suited for the analysis of long biopolymers, as has already been shown for DNA sequencing. However, proteins, unlike DNA molecules, are not uniformly charged and harbor complex tertiary structures. Consequently, the ability of nanopores to analyze unfolded full-length proteins has remained elusive. Here, we evaluate the use of heat denaturation and the anionic surfactant sodium dodecyl sulfate (SDS) to facilitate electrokinetic nanopore sensing of unfolded proteins. Specifically, we characterize the voltage dependence translocation dynamics of a wide molecular weight range of proteins (from 14 to 130 kDa) through sub-5 nm solid-state nanopores, using a SDS concentration below the critical micelle concentration. Our results suggest that proteins' translocation dynamics are significantly slower than expected, presumably due to the smaller nanopore diameters used in our study and the role of the electroosmotic force opposing the translocation direction. This allows us to distinguish among the proteins of different molecular weights based on their dwell time and electrical charge deficit. Given the simplicity of the protein denaturation assay and circumvention of the tailor-made necessities for sensing protein of different folded sizes, shapes, and charges, this approach can facilitate the development of a whole proteome identification technique.

Salt Gradient Control of Translocation Dynamics in a Solid-State Nanopore

Tuning capture rates and translocation time of analytes in solid-state nanopores are one of the major challenges for their use in detecting and analyzing individual nanoscale objects via ionic current measurements. Here, we report on the use of salt gradient for the fine control of capture-to-translocation dynamics in 300 nm sized SiN_x nanopores. We demonstrated a decrease up to a factor of 3 in the electrophoretic speed of nanoparticles at the pore exit along with an over 3-fold increase in particle detection efficiency by subjecting a 5-fold ion concentration difference across the dielectric membrane. The improvement in the sensor performance was elucidated to be a result of the salt-gradient-mediated electric field and electroosmotic flow asymmetry at nanochannel orifices. The present findings can be used to enhance nanopore sensing capability for detecting biomolecules such as amyloids and proteins.

Wetting of nanopores probed with pressure

Nanopores are both a tool to study single-molecule biophysics and nanoscale ion transport, but also a promising material for desalination or osmotic power generation. Understanding the physics underlying ion transport through nano-sized pores allows better design of porous membrane materials. Material surfaces can present hydrophobicity, a property which can make them prone to formation of surface nanobubbles. Nanobubbles can influence the electrical transport properties of such devices. We demonstrate an approach which uses hydraulic pressure to probe the electrical transport properties of solid state nanopores. We show how pressure can be used to wet pores, and how it allows control over bubbles or other contaminants in the nanometer scale range normally unachievable using only an electrical driving force. Molybdenum disulfide is then used as a typical example of a 2D material on which we demonstrate wetting and bubble induced nonlinear and linear conductance in the regimes typically used with these experiments. We show that by using pressure one can identify and evade wetting artifacts.

Transport of neutral and charged nanorods across varying-section channels

We study the dynamics of neutral and charged rods embedded in varying-section channels. By means of systematic approximations, we derive the dependence of the local diffusion coefficient on both the geometry and charge of the rods. This microscopic insight allows us to provide predictions for the permeability of varying-section channels to rods with diverse lengths, aspect ratios and charge. Our analysis shows that the dynamics of charged rods is sensitive to the geometry of the channel and that their transport can be controlled by tuning both the shape of the confining walls and the charge of the rod. Interestingly, we find that the channel permeability does not depend monotonically on the charge of the rod. This opens the possibility of a novel mechanism to separate charged rods.

Dielectric Coatings for Resistive Pulse Sensing Using Solid-State Pores

The present study reports on the systematic characterization of the effectiveness of dielectric coating to tailor capture-to-translocation dynamics of single particles in solid-state pores. We covered the surface of SiN_x membranes with SiO₂, HfO₂, Al₂O₃, TiO₂, or ZnO, which allowed us to change the ζ-potential at the pore wall, reflecting the isoelectric points of these coating materials. Resistive pulse measurements of negatively charged polystyrene beads elucidated more facile electrophoretic capture of the particles and slower translocation motions in the channel under more negative electric potential at the oxide surface. These findings provide a guide to engineer pore wall surface for optimizing the translocation dynamics for efficient sensing of particles and molecules.

Analytical Model for Particle Capture in Nanopores Elucidates Competition among Electrophoresis, Electroosmosis, and Dielectrophoresis

The interaction between nanoparticles dispersed in a fluid and nanopores is governed by the interplay of hydrodynamical, electrical, and chemical effects. We developed a theory for particle capture in nanopores and derived analytical expressions for the capture rate under the concurrent action of electrical forces, fluid advection, and Brownian motion. Our approach naturally splits the average capture time in two terms, an approaching time due to the migration of particles from the bulk to the pore mouth and an entrance time associated with a free-energy barrier at the pore entrance. Within this theoretical framework, we described the standard experimental condition where a particle concentration is driven into the pore by an applied voltage, with specific focus on different capture mechanisms: under pure electrophoretic force, in the presence of a competition between electrophoresis and electroosmosis, and finally under dielectrophoretic reorientation of dipolar particles. Our theory predicts that dielectrophoresis is able to induce capture for both positive and negative voltages. We performed a dedicated experiment involving a biological nanopore (α-hemolysin) and a rigid dipolar dumbbell (realized with a β-hairpin peptide) that confirms the theoretically proposed capture mechanism.

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.

Modeling the Effective Conductance Drop Due to a Particle in a Solid State Nanopore Towards Optimized Design

An understanding of the current change in a solid state nanopore due to particle movement or capture is crucial for improvement of nanopore based sensing technologies. For lower aspect ratio pores, which are gaining importance due to their high sensitivity, there is interplay between access and pore resistance and the existing theories for computation of access resistance cannot explain most of the experimental observations. Hence, there is a need to develop a comprehensive model for calculating the effective conductance drop in presence of particles in a solid state nanopore. In this paper, we develop analytical models to calculate both the access and pore resistance in presence of particle at different positions during translocation and also when captured by receptors in functionalized nanopores. A wide range of pore geometry and molar strength has been investigated. Taking into consideration the positional uncertainty during particle translocation, the effective resistance sensitivity has been found to agree very well with the experimental observations in low aspect ratio pore. Additionally, we observe that in functionalized nanopores, a pore of higher diameter results in around 50% increase in sensitivity compared to a pore with half its diameter, which indicates the scope of design optimization in such systems.

Surface coatings for solid-state nanopores

Since their introduction in 2001, solid-state nanopores have been increasingly exploited for the detection and characterization of biomolecules ranging from single DNA strands to protein complexes. A major factor that enables the application of nanopores to the analysis and characterization of a broad range of macromolecules is the preparation of coatings on the pore wall to either prevent non-specific adhesion of molecules or to facilitate specific interactions of molecules of interest within the pore. Surface coatings can therefore be useful to minimize clogging of nanopores or to increase the residence time of target analytes in the pore. This review article describes various coatings and their utility for changing pore diameters, increasing the stability of nanopores, reducing non-specific interactions, manipulating surface charges, enabling interactions with specific target molecules, and reducing the noise of current recordings through nanopores. We compare the coating methods with respect to the ease of preparing the coating, the stability of the coating and the requirement for specialized equipment to prepare the coating.

Entrance effects in concentration-gradient-driven flow through an ultrathin porous membrane

Transport of liquid mixtures through porous membranes is central to processes such as desalination, chemical separations, and energy harvesting, with ultrathin membranes made from novel 2D nanomaterials showing exceptional promise. Here, we derive, for the first time, general equations for the solution and solute fluxes through a circular pore in an ultrathin planar membrane induced by a solute concentration gradient. We show that the equations accurately capture the fluid fluxes measured in finite-element numerical simulations for weak solute-membrane interactions. We also derive scaling laws for these fluxes as a function of the pore size and the strength and range of solute-membrane interactions. These scaling relationships differ markedly from those for concentration-gradient-driven flow through a long cylindrical pore or for flow induced by a pressure gradient or an electric field through a pore in an ultrathin membrane. These results have broad implications for transport of liquid mixtures through membranes with thickness on the order of the characteristic pore size.

Non-Negligible Roles of Pore Size Distribution on Electroosmotic Flow in Nanoporous Materials

Electroosmotic flow in nanoporous materials is of fundamental importance for the design and development of filtration membranes and electrochemical devices such as supercapacitors and batteries. Recent experiments suggest that ion transport in a porous network is substantially different from that in individual nanochannels due to the pore size distribution and pore connectivity. Herein, we report a theoretical framework for ion transport in nanoporous materials by combing the classical density functional theory to describe the electrical double layer (EDL) structure, the Navier-Stokes equation for the fluid flow, and the effective medium approximation to bridge the gap between individual nanopores and the network connectivity. We find that ion conductivity in nanoporous materials is extremely sensitive to the pore size distribution when the average size of micropores is comparable to the EDL thickness. The theoretical predictions provide an explanation of the giant gap between the conductivity of a single pore and that of a porous network and highlight the mechanism of ion transport through nanoporous materials important for numerous practical applications.

Osmosis, from molecular insights to large-scale applications

Osmosis is a universal phenomenon occurring in a broad variety of processes and fields. It is the archetype of entropic forces, both trivial in its fundamental expression - the van 't Hoff perfect gas law - and highly subtle in its physical roots. While osmosis is intimately linked with transport across membranes, it also manifests itself as an interfacial transport phenomenon: the so-called diffusio-osmosis and -phoresis, whose consequences are presently actively explored for example for the manipulation of colloidal suspensions or the development of active colloidal swimmers. Here we give a global and unifying view of the phenomenon of osmosis and its consequences with a multi-disciplinary perspective. Pushing the fundamental understanding of osmosis allows one to propose new perspectives for different fields and we highlight a number of examples along these lines, for example introducing the concepts of osmotic diodes, active separation and far from equilibrium osmosis, raising in turn fundamental questions in the thermodynamics of separation. The applications of osmosis are also obviously considerable and span very diverse fields. Here we discuss a selection of phenomena and applications where osmosis shows great promises: osmotic phenomena in membrane science (with recent developments in separation, desalination, reverse osmosis for water purification thanks in particular to the emergence of new nanomaterials); applications in biology and health (in particular discussing the kidney filtration process); osmosis and energy harvesting (in particular, osmotic power and blue energy as well as capacitive mixing); applications in detergency and cleaning, as well as for oil recovery in porous media.

Salt-Mediated Nanopore Detection of ADAM-17

ADAM-17 (a disintegrin and metalloproteinase 17) plays an important role in various physiological and pathophysiological processes. Overexpression/underexpression of ADAM-17 could lead to various diseases. In this work, by taking advantage of ionic strength and salt gradient, and monitoring the cleavage of a substrate peptide by ADAM-17 in a nanopore, we developed a label-free sensor for the rapid detection of ADAM-17. The sensor was highly sensitive and selective: picomolar concentrations of ADAM-17 could be detected within minutes, while structure similar proteases such as ADAM-9 and MMP-9 did not interfere with its detection. Our developed nanopore sensing strategy should find useful applications in the development of nanopore sensors for other proteases of biological, pharmaceutical, and medical importance.

Brownian dynamics simulations of the ionic current traces for a neutral nanoparticle translocating through a nanopore

In this work, the ionic current blockades due to the translocation of a neutral spherical nanoparticle through a nanopore in a solid state membrane are computed. We use a Brownian dynamics approach, in conjunction with a full three-dimensional self-consistent solution of the Poisson-Nernst-Planck and Navier-Stockes system of equations to describe realistic ionic current response arising due to the random motion of a nanoparticle through a nanopore. We find that in addition to the usual geometric blockade, the variations of the current along the axis of the pore are largely caused by a concentration polarization induced by the presence of the translocating nanoparticle in the nanopore while the current changes in the radial (perpendicular to the axis) direction occur because of the local build up of the ionic charge between the particle and the nanopore surface. By performing statistical analysis of the current traces, we also observe that, in general, smaller current blockade values correspond to faster translocation times, while increased dwell times result in a larger current decrease.

TEM Tomography of Pores with Application to Computational Nanoscale Flows in Nanoporous Silicon Nitride (NPN)

Silicon nanomembrane technologies (NPN, pnc-Si, and others) have been used commercially as electron microscopy (EM) substrates, and as filters with nanometer-resolution size cut-offs. Combined with EM, these materials provide a platform for catching or suspending nanoscale-size structures for analysis. Usefully, the nanomembrane itself can be manufactured to achieve a variety of nanopore topographies. The size, shapes, and surfaces of nanopores will influence transport, fouling, sieving, and electrical behavior. Electron tomography (ET) techniques used to recreate nanoscale-sized structures would provide an excellent way to capture this variation. Therefore, we modified a sample holder to accept our standardized 5.4 mm × 5.4 mm silicon nanomembrane chips and imaged NPN nanomembranes (50–100 nm thick, 10–100 nm nanopore diameters) using transmission electron microscopy (TEM). After imaging and ET reconstruction using a series of freely available tools (ImageJ, TomoJ, SEG3D2, Meshlab), we used COMSOL Multiphysics™ to simulate fluid flow inside a reconstructed nanopore. The results show flow profiles with significantly more complexity than a simple cylindrical model would predict, with regions of stagnation inside the nanopores. We expect that such tomographic reconstructions of ultrathin nanopores will be valuable in elucidating the physics that underlie the many applications of silicon nanomembranes.

Local pressure for confined systems

We derive a general closed expression for the local pressure exerted onto the corrugated walls of a channel confining a fluid medium. When the fluid medium is at equilibrium, the local pressure is a functional of the shape of the walls. It is shown that, due to the intrinsic nonlocal character of the interactions among the particles forming the fluid, the applicability of approximate schemes such as the concept of a surface of tension or morphometric thermodynamics is limited to wall curvatures that are small compared to the range of particle-particle interactions.