Hybrid continuum-atomistic approach to model electrokinetics in nanofluidics

In this study, for the first time, a hybrid continuum-atomistic based model is proposed for electrokinetics, electroosmosis and electrophoresis, through nanochannels. Although continuum based methods are accurate enough to model fluid flow and electric potential in nanofluidics (in dimensions larger than 4 nm), ionic concentration is too low in nanochannels for the continuum assumption to be valid. On the other hand, the non-continuum based approaches are too time-consuming and therefore is limited to simple geometries, in practice. Here, to propose an efficient hybrid continuum-atomistic method of modelling the electrokinetics in nanochannels; the fluid flow and electric potential are computed based on continuum hypothesis coupled with an atomistic Lagrangian approach for the ionic transport. The results of the model are compared to and validated by the results of the molecular dynamics technique for a couple of case studies. Then, the influences of bulk ionic concentration, external electric field, size of nanochannel, and surface electric charge on the electrokinetic flow and ionic mass transfer are investigated, carefully. The hybrid continuum-atomistic method is a promising approach to model more complicated geometries and investigate more details of the electrokinetics in nanofluidics.     

Diffusioosmotic flows in slit nanochannels

Diffusioosmotic flows of electrolyte solutions in slit nanochannels with homogeneous surface charges induced by electrolyte concentration gradients in the absence of externally applied pressure gradients and potential differences are investigated theoretically. A continuum mathematical model consisting of the strongly coupled Nernst-Planck equations for the ionic species' concentrations, the Poisson equation for the electric potential in the electrolyte solution, and the Navier-Stokes equations for the flow field is numerically solved simultaneously. The induced diffusioosmotic flow through the nanochannel is computed as functions of the externally imposed concentration gradient, the concentration of the electrolyte solution, and the surface charge density along the walls of the nanochannel. With the externally applied electrolyte concentration gradient, a strongly spatially dependent electric field and pressure gradient are induced within the nanochannel that, in turn, generate a spatially dependent diffusioosmotic flow. The diffusioosmotic flow is opposite to the applied concentration gradient for a relatively low bulk electrolyte concentration. However, the electrolyte solution flows from one end of the nanochannel with a higher electrolyte concentration to the other end with a lower electrolyte concentration when the bulk electrolyte concentration is relatively high. There is an optimal concentration gradient under which the flow rate attains the maximum. The induced flow is enhanced with the increase in the fixed surface charge along the wall of the nanochannel for a relatively low bulk electrolyte concentration.     

Electrokinetic flow and electric conduction of salt-free solutions in a capillary

The electrokinetic flow and accompanied electric conduction of a salt-free solution in the axial direction of a charged circular capillary are analyzed. No assumptions are made about the surface charge density (or surface potential) and electrokinetic radius of the capillary, which are interrelated. The Poisson–Boltzmann equation and modified Navier–Stokes equation are solved for the electrostatic potential distribution and fluid velocity profile, respectively. Closed-form formulas for the electroosmotic mobility and electric conductivity in the capillary are derived in terms of the surface charge density. The relative surface potential, electroosmotic mobility, and electric conductivity are monotonic increasing functions of the surface charge density and electrokinetic radius. However, the rises of the relative surface potential and electroosmotic mobility with an increase in the surface charge density are suppressed substantially when it is high due to the effect of counterion condensation. The analytical prediction that the electroosmotic mobility grows with increases in the surface charge density and electrokinetic radius agrees with the experimental results for salt-free solutions in circular microchannels in the literature.     

Predicting ion concentration polarization and analyte stacking/focusing at nanofluidic interfaces

We report on the investigation of electropreconcentration phenomena in micro-/nanofluidic devices integrating 100 μm long nanochannels using 2D COMSOL simulations based on the coupled Poisson–Nernst–Planck and Navier–Stokes system of equations. Our numerical model is used to demonstrate the influence of key governing parameters such as electrolyte concentration, surface charge density, and applied axial electric field on ion concentration polarization (ICP) dynamics in our system. Under sufficiently extreme surface-charge-governed transport conditions, ICP propagation is shown to enable various transient and stationary stacking and counter-flow gradient focusing mechanisms of anionic analytes. We resolve these spatiotemporal dynamics of analytes and electrolyte ICP over disparate time and length scales, and confirm previous findings that the greatest enhancement is observed when a system is tuned for analyte focusing at the charge, excluding microchannel, nanochannel electrical double layer (EDL) interface. Moreover, we demonstrate that such tuning can readily be achieved by including additional nanochannels oriented parallel to the electric field between two microchannels, effectively increasing the overall perm-selectivity and leading to enhanced focusing at the EDL interfaces. This approach shows promise in providing added control over the extent of ICP in electrokinetic systems, particularly under circumstances in which relatively weak ICP effects are observed using only a single channel.     

Electrokinetic ion transport in confined micro‐nanochannel

In this paper, a confined micronanochannel is presented to concentrate ions in a restricted zone. A general model exploiting the Poisson–Nernst–Plank equations coupled with the Navier–Stokes equation is employed to simulate the electrokinetic ion transport. The influences of the micronanochannel dimension and the surface charge density on the potential distribution, the ion concentration, and the fluid flow are investigated. The numerical results show that the potential drop depends mainly on the nanochannel, instead of the confined channel. Both decreasing the width and increasing the length enhance the ion enrichment performance. For a given nanochannel, ultimate value of ion concentration may be determined by the potential at the center point of the nanochannel. The study also shows that the enrichment stability can be improved by increasing the micronanochannel width, decreasing the micronanochannel length and reducing the surface charge density.     

Mass transport in nanofluidic devices

Nanofluidics is a recent appearing research field, introduced in 1995 as an analogue of the field of microfluidics, and has been becoming popular in the past few years. The proximity of the channel dimension, the Debye length, and the size of biomolecules such as DNA and proteins gives the unique features of nanofluidic devices. Of various unique properties of the nanofluidics, mass transport in nanochannel plays determining roles in fundamental reaches and practical applications of nanofluidic device. Thus, much work including numerical and experimental researches has been performed to investigate the mass transport behaviors in nanofluidic devices. This review summarizes the fabrication technologies for nanofluidic devices, the mass transport behaviors in nanochannel, and their applications in bioanalysis. The main focus will be laid on the effects of nanochannel size and surface charge on mass transport including electrokinetic transport of charged analytes, diffusion of electric neutral molecules, ionic current rectification, concentration polarization, nonlinear electrokinetic flow at the micro-nanofluidic interfaces.     

Electrokinetic transport through the nanopores in cell membrane during electroporation

In electroporation, applied electric field creates hydrophilic nanopores in a cell membrane that can serve as a pathway for inserting biological samples to the cell. It is highly desirable to understand the ionic transfer and fluid flow through the nanopores in order to control and improve the cell transfection. Because of submicron dimensions, conventional theories of electrokinetics may lose their applicability in such nanopores. In the current study, the Poisson-Nernst-Planck equations along with modified Navier-Stokes equations and the continuity equation are solved in order to find electric potential, fluid flow, and ionic concentration through the nanopores. The results show that the electric potential, velocity field, and ionic concentration vary with the size of the nanopores and are different through the nanopores located at the front and backside of the cell membrane. However, on a given side of the cell membrane, angular position of nanopores has fewer influences on liquid flow and ionic transfer. By increasing the radius of the nanopores, the averaged velocity and ionic concentration through the nanopores are increased. It is also shown that, in the presence of electric pulse, electrokinetic effects (electroosmosis and electrophoresis) have significant influences on ionic mass transfer through the nanopores, while the effect of diffusion on ionic mass flux is negligible in comparison with electrokinetics. Increasing the radius of the nanopores intensifies the effect of convection (electroosmosis) in comparison with electrophoresis on ionic flux.     

Induced electrokinetic transport in micro-nanofluidic interconnect devices

Hybrid micro-nanofluidic interconnect devices can be used to control analyte transfer from one microchannel to the other through a nanochannel under rest, injection, and recovery stages of operation by varying the applied potential bias. Using numerical simulations based on coupled transient Poisson-Nernst-Planck and Stokes equations, we examine the electrokinetic transport in a gateable device consisting of two 100 microm long, 1 microm wide negatively charged microchannels connected by a 1 microm long, 10 nm wide positively charged nanochannel under both positive and negative bias potentials. During injection, accumulation of ions is observed at the micro-nano interface region with the positive potential and depletion of ions is observed at the other micro-nano junction region. Net space charge in the depletion region gives rise to nonlinear electrokinetic transport during the recovery stage due to induced pressure, induced electroosmotic flow of the second kind, and complex flow circulations. Ionic currents are computed as a function of time for both positive and negative bias potentials for the three stages. Analytical expressions derived for ion current variation are in agreement with the simulated results. In the presence of multiple accumulation or depletion regions, we show that a hybrid micro-nano device can be designed to function as a logic gate.     

Solute transport and separation in nanochannel chromatography

An analytical model is developed to study the solute transport and separation in pressure-driven liquid flow through cylindrical nanochannels. The flow-induced streaming potential is found to significantly affect the solute speed, retention and dispersion in nanochannel chromatography. These effects are sensitive to the solute charge, and found to be mainly dependent on an electrokinetic "figure of merit".     

Scaling of electrokinetic transport in nanometer channels

Electrokinetic transport is a popular transport mechanism used in nanofluidic systems, and understanding its scaling behavior is important for the design and optimization of nanofluidic devices. In this article, we report on the scaling of electroosmotic flow and ionic conductivity in positively charged slit nanochannels by using continuum and atomistic simulations. The effects of confinement and surface charge are discussed in detail. In particular, we found that the viscosity of the interfacial water increases substantially as the surface charge density increases and the electrophoretic mobility of the interfacial ions decreases. We show that such effects can influence the scaling of the electrokinetic transport in confined nanochannels significantly.     

Modeling electrokinetics in ionic liquids

Using direct numerical simulations, we provide a thorough study regarding the electrokinetics of ionic liquids. In particular, modified Poisson–Nernst–Planck equations are solved to capture the crowding and overscreening effects characteristic of an ionic liquid. For modeling electrokinetic flows in an ionic liquid, the modified Poisson‐Nernst‐Planck equations are coupled with Navier–Stokes equations to study the coupling of ion transport, hydrodynamics, and electrostatic forces. Specifically, we consider the ion transport between two parallel charged surfaces, charging dynamics in a nanopore, capacitance of electric double‐layer capacitors, electroosmotic flow in a nanochannel, electroconvective instability on a plane ion‐selective surface, and electroconvective flow on a curved ion‐selective surface. We also discuss how crowding and overscreening and their interplay affect the electrokinetic behaviors of ionic liquids in these application problems.     

纳通道的物质传输特性及应用

近年来,随着材料科学、微纳加工技术和微纳尺度物质传输理论的发展,纳通道技术得到了越来越多的研究和关注。纳通道包括生物纳通道和人工纳通道,其孔径通常为1~100 nm。在这一尺度下,通道表面与通道内物质之间的作用概率大大增强,使得纳通道表现出许多与宏观体系不同的物质传输特性,例如通道表面电荷与通道内离子之间的静电作用产生了离子选择性,通道内电化学势的不对称分布产生了离子整流特性,物质传输过程中占据通道产生了阻塞脉冲特性等。纳通道中的这些物质传输特性在传感、分离、能源等领域具有广泛应用,例如通过对纳通道进行功能化修饰可以实现门控离子传输;利用亚纳米尺度的通道可以实现单分子传感;利用通道与传输物质之间的相互作用可以实现离子、分子、纳米粒子的分离;利用纳通道的离子选择性可以在通道内实现电荷分离,将不同形式的能量（如光、热、压力、盐差等）高效转化为电能。纳通道技术是化学、材料科学、纳米技术等多学科的交叉集合,在解决生物、环境、能源等基本问题方面具有良好的前景。该文综述了近10年来与纳通道物质传输理论以及纳通道技术应用相关的前沿研究,梳理了纳通道技术的发展过程,并对其在各个领域的应用进行了总结与展望。     

Electrokinetics in nanochannels: part I. Electric double layer overlap and channel-to-well equilibrium

In this paper a new model is described for calculating the electric potential field in a long, thin nanochannel with overlapped electric double layers. Electrolyte concentration in the nanochannel is predicted self-consistently via equilibrium between ionic solution in the wells and within the nanochannel. Differently than published models that require detailed iterative numerical solutions of coupled differential equations, the framework presented here is self-consistent and predictions are obtained solving a simple one-dimensional integral. The derivation clearly shows that the electric potential field depends on three new parameters: the ratio of ion density in the channel to ion density in the wells; the ratio of free-charge density to bulk ion density within the channel; and a modified Debye-Hückel thickness, which is the relevant scale for shielding of surface net charge. For completeness, three wall-surface boundary conditions are analyzed: specified zeta-potential; specified surface net charge density; and charge regulation. Predictions of experimentally observable quantities based on the model proposed here, such as depth-averaged electroosmotic flow and net ionic current, are significantly different than results from previous overlapped electric double layer models. In this first paper of a series of two, predictions are presented where channel depth is varied at constant well concentration. Results show that under conditions of electric double layer overlap, electroosmosis contributes only a small fraction of the net ionic current, and that most of the measurable current is due to ionic conduction in conditions of increased counterion density in the nanochannel. In the second of this two-paper series, predictions are presented where well-concentration is varied and the channel depth is held constant, and the model described here is employed to study the dependence of ion mobility on ionic strength, and compare predictions to measurements of ionic current as a function of channel depth and ion density.     

A nanochannel array based device for determination of the isoelectric point of confined proteins

A nanochannel array based nanodevice can mimic the biological environments and thus unveil the natural properties, conformation and recognition information of biomolecules such as proteins and DNA in confined spaces. Here we report that porous anodic alumina (PAA) of a highly parallel nanochannel array covalently modified with proteins significantly modulates the transport of a negatively charged probe of ferricyanide due to the electrostatic interactions between the probes and modified nanochannel inner surface. Results show that such electrostatic interaction exists in a wide range of ionic strength from 1 mM to 100 mM in 20 nm nanochannels modified with proteins (hemoglobin, bovine serum albumin, and goat anti-rabbit IgG secondary antibody). In addition, the maximal steady-state flux of the charged probe through the modified nanochannel array is directly related to the ionic strength which determines the electric double layer thickness and solution pH which modulates the nanochannel surface charge. Thus, the modulated mass transport of the probe by solution pH can be used to study the charge properties of the immobilized proteins in nanochannel confined conditions, leading us to obtain the isoelectric point (pI) of the proteins confined in nanochannels. The determined pI values of two known proteins of hemoglobin and bovine serum albumin are close to the ones of the same proteins covalently modified on a 3-mercaptopropionic acid self-assembled monolayer/gold electrode. In addition, the pI of an unknown protein of goat anti-rabbit IgG secondary antibody confined in nanochannels was determined to be 6.3. Finally, the confinement effect of nanochannels on the charge properties of immobilized proteins has been discussed.     

Influence of the three-dimensional heterogeneous roughness on electrokinetic transport in microchannels

Surface roughness has been considered as a passive means of enhancing species mixing in electroosmotic flow through microfluidic systems. It is highly desirable to understand the synergetic effect of three-dimensional (3D) roughness and surface heterogeneity on the electrokinetic flow through microchannels. In this study, we developed a three-dimensional finite-volume-based numerical model to simulate electroosmotic transport in a slit microchannel (formed between two parallel plates) with numerous heterogeneous prismatic roughness elements arranged symmetrically and asymmetrically on the microchannel walls. We consider that all 3D prismatic rough elements have the same surface charge or zeta potential, the substrate (the microchannel wall) surface has a different zeta potential. The results showed that the rough channel's geometry and the electroosmotic mobility ratio of the roughness elements' surface to that of the substrate, epsilon(mu), have a dramatic influence on the induced-pressure field, the electroosmotic flow patterns, and the electroosmotic flow rate in the heterogeneous rough microchannels. The associated sample-species transport presents a tidal-wave-like concentration field at the intersection between four neighboring rough elements under low epsilon(mu) values and has a concentration field similar to that of the smooth channels under high epsilon(mu) values.     

Streaming potential and electroosmotic flow in heterogeneous circular microchannels with nonuniform zeta potentials: requirements of flow rate and current continuities

Real surfaces are typically heterogeneous, and microchannels with heterogeneous surfaces are commonly found due to fabrication defects, material impurities, and chemical adsorption from solution. Such surface heterogeneity causes a nonuniform surface potential along the microchannel. Other than surface heterogeneity, one could also pattern the various surface potentials along the microchannels. To understand how such variations affect electrokinetic flow, we proposed a model to describe its behavior in circular microchannels with nonuniform surface potentials. Unlike other models, we considered the continuities of flow rate and electric current simultaneously. These requirements cause a nonuniform electric field distribution and pressure gradient along the channel for both pressure-driven flow (streaming potential) and electric-field-driven flow (electroosmosis). The induced nonuniform pressure and electric field influence the electrokinetic flow in terms of the velocity profile, the flow rate, and the streaming potential.     

Electrokinetic transport and separations in fluidic nanochannels

This article presents a summary of theory, experimental studies, and results for the electrokinetic transport in small fluidic nanochannels. The main focus is on the effect of the electric double layer on the EOF, electric current, and electrophoresis of charged analytes. The double layer thickness can be of the same order as the width of the nanochannels, which has an impact on the transport by shaping the fluid velocity profile, local distributions of the electrolytes, and charged analytes. Our theoretical consideration is limited to continuum analysis where the equations of classical hydrodynamics and electrodynamics still apply. We show that small channels may lead to qualitatively new effects like selective ionic transport based on charge number as well as different modes for molecular separation. These new possibilities together with the rapid development of nanofabrication capabilities lead to an extensive experimental effort to utilize nanochannels for a variety of applications, which are also discussed and analyzed in this review.     

Diffusiophoresis in a suspension of charge-regulating colloidal spheres

An analytical study of diffusiophoresis in a homogeneous suspension of identical spherical charge-regulating particles with an arbitrary thickness of the electric double layers in a solution of a symmetrically charged electrolyte with a uniform prescribed concentration gradient is presented. The charge regulation due to association/dissociation reactions of ionogenic functional groups on the particle surface is approximated by a linearized regulation model, which specifies a linear relationship between the surface charge density and the surface potential. The effects of particle-particle electrohydrodynamic interactions are taken into account by employing a unit cell model, and the overlap of the double layers of adjacent particles is allowed. The electrokinetic equations that govern the electric potential profile, the ionic concentration distributions, and the fluid flow field in the electrolyte solution surrounding the particle in a unit cell are linearized assuming that the system is only slightly distorted from equilibrium. Using a regular perturbation method, these linearized equations are solved with the equilibrium surface charge density (or zeta potential) of the particle as the small perturbation parameter. Closed-form formulas for the diffusiophoretic velocity of the charge-regulating sphere correct to the second order of its surface charge density or zeta potential are derived. Our results indicate that the charge regulation effect on the diffusiophoretic mobility is quite sensitive to the boundary condition for the electric potential specified at the outer surface of the unit cell. For the limiting cases of a very dilute suspension and a very thin or very thick electric double layer, the particle velocity is independent of the charge regulation parameter.     

Diffusioosmosis of electrolyte solutions in a fine capillary slit

The steady diffusioosmotic flows of an electrolyte solution along a charged plane wall and in a capillary channel between two identical parallel charged plates generated by an imposed tangential concentration gradient are theoretically investigated. The plane walls may have either a constant surface potential or a constant surface charge density. The electrical double layers adjacent to the charged walls may have an arbitrary thickness and their electrostatic potential distributions are determined by the Poisson-Boltzmann equation. Solving a modified Navier-Stokes equation with the constraint of no net electric current arising from the cocurrent diffusion, electric migration, and diffusioosmotic convection of the electrolyte ions, the macroscopic electric field and the fluid velocity along the tangential direction induced by the imposed electrolyte concentration gradient are obtained semianalytically as a function of the lateral position in a self-consistent way. The direction of the diffusioosmotic flow relative to the concentration gradient is determined by the combination of the zeta potential (or surface charge density) of the wall, the properties of the electrolyte solution, and other relevant factors. For a given concentration gradient of an electrolyte along a plane wall, the magnitude of fluid velocity at a position in general increases with an increase in its electrokinetic distance from the wall, but there are exceptions. The effect of the lateral distribution of the induced tangential electric field and the relaxation effect in the double layer on the diffusioosmotic flow are found to be very significant.     

Electrokinetic flow in a capillary with a charge-regulating surface polymer layer

An analytical study of the steady electrokinetic flow in a long uniform capillary tube or slit is presented. The inside wall of the capillary is covered by a layer of adsorbed or covalently bound charge-regulating polymer in equilibrium with the ambient electrolyte solution. In this solvent-permeable and ion-penetrable surface polyelectrolyte layer, ionogenic functional groups and frictional segments are assumed to distribute at uniform densities. The electrical potential and space charge density distributions in the cross section of the capillary are obtained by solving the linearized Poisson-Boltzmann equation. The fluid velocity profile due to the application of an electric field and a pressure gradient through the capillary is obtained from the analytical solution of a modified Navier-Stokes/Brinkman equation. Explicit formulas for the electroosmotic velocity, the average fluid velocity and electric current density on the cross section, and the streaming potential in the capillary are also derived. The results demonstrate that the direction of the electroosmotic flow and the magnitudes of the fluid velocity and electric current density are dominated by the fixed charge density inside the surface polymer layer, which is determined by the regulation characteristics such as the dissociation equilibrium constants of the ionogenic functional groups in the surface layer and the concentration of the potential-determining ions in the bulk solution.