Electrokinetics in nanochannels: part II. Mobility dependence on ion density and ionic current measurements

In the first of this two-paper series, a new model was developed for calculating the electric potential field in a long, thin nanochannel with overlapped electric double layers. The model takes into account the dependence of ion mobility on local ion densities and pH. This model is used here to study and demonstrate the effect of ion density and pH on ionic current measurements. A comparison is shown of predictions based on each of three boundary conditions, as studied in Part I. The model developed in Part I is validated by comparing simulations with measurements of ionic current as a function of sodium borate concentration. Results show that predictions based on extended Debye-Hückel theory for ion mobility significantly improve the accuracy of simulations, but that these do not predict exact scaling behavior. A simple bulk conductivity measurement used as input parameter for the simulations, in place of the predicted bulk conductivity (K(0)), guarantees agreement with data in the thin EDL region. Results also indicate that the charge regulation boundary condition, complemented with an adequate bulk electrolyte model, provides better agreement with experimental trends than the specified zeta potential or specified surface net charge boundary conditions. Further, it is shown that currents due to advection (by electroosmotic flow) are in all cases studied less than 25% of the total current in the system.     

Effect of gate length and dielectric thickness on ion and fluid transport in a fluidic nanochannel

We have simulated the effect of gate length and dielectric thickness on ion and fluid transport in a fluidic nanochannel with negative surface charge on its walls. A short gate is unable to induce significant cation enrichment in the nanochannel and ion current is controlled mostly by cation depletion at positive gate potentials. The cation enrichment increases with increasing gate length and/or decreasing dielectric thickness due to higher changes induced in the surface charge density and zeta-potential. Thus, long gates and thin dielectric layers are more effective in controlling ion current. The model without Navier-Stokes equations is unable to correctly predict phenomena such as cation enrichment, increase in channel conductivity, and decreasing electric field. Body force and induced fluid velocity decrease slowly and then rapidly with gate potentials. The effectiveness of ion current control by a gate reduces with increasing surface charge density due to reduced fractional change in zeta-potential.     

Ion concentration polarization near microchannel-nanochannel interfaces: effect of pH value

This study investigates the effect of the pH value on the ion concentration polarization phenomenon and the nonlinear current-voltage characteristics of a hybrid soda-lime glass micro/nanochannel for a constant KCl salt concentration of about 1 mM. The experimental results show that the electrical conductance of the nanochannel in the Ohmic regime and the critical threshold voltage of the limiting current are both dependent on the pH value of the salt solution when the electrical double layer thickness is considerable in the nanochannel. Specifically, the nanochannel conductance increases and the critical threshold voltage for the limiting current decreases as the pH value is increased. It also suggests that a higher pH value induces a higher surface charge density on the nanochannel walls, and therefore increases both the ionic conductance and the counter-ion flux within the nanochannel.     

A quantitative model to evaluate the ion-enrichment and ion-depletion effect at microchannel-nanochannel junctions

In a nanometer-scale fluidic channel (nanochannel), coions are depleted while counterions are concentrated due to the electric double layer (EDL) overlap. When an electric field is applied across the nanochannel, ions are enriched at one end and depleted at the other end of the nanochannel. This phenomenon is termed the ion-enrichment and ion-epletion (IEID) effect. In this paper, we develop a theoretical model to evaluate this effect. The model takes into accounts not only the biased electrophoretic migrations but also the net charge transportation caused by electroosmotic flow. In addition, we consider the conductance change inside the nanochannel in assessing the electric field strength across it. We employ our recently developed protocol to measure these values. We establish a protocol to measure/quantitate the IEID effect. Finally, we compare the calculated results with the experimentally measured data and show good agreements between them.     

Electrokinetic transport through nanochannels

This article presents a numerical study of the electrokinetic transport phenomena (electroosmosis and electrophoresis) in a three-dimensional nanochannel with a circular cross-section. Due to the nanometer dimensions, the Boltzmann distribution of the ions is not valid in the nanochannels. Therefore, the conventional theories of electrokinetic flow through the microchannels such as Poisson-Boltzmann equation and Helmholtz-Smoluchowski slip velocity approach are no longer applicable. In the current study, a set of coupled partial differential equations including Poisson-Nernst-Plank equation, Navier-Stokes, and continuity equations is solved to find the electric potential field, ionic concentration field, and the velocity field in the three-dimensional nanochannel. The effects of surface electric charge and the radius of nanochannel on the electric potential, liquid flow, and ionic transport are investigated. Unlike the microchannels, the electric potential field, ionic concentration field, and velocity field are strongly size-dependent in nanochannels. The electric potential gradient along the nanochannel also depends on the surface electric charge of the nanochannel. More counter ions than the coions are transported through the nanochannel. The ionic concentration enrichment at the entrance and the exit of the nanochannel is completely evident from the simulation results. The study also shows that the flow velocity in the nanochannel is higher when the surface electric charge is stronger or the radius of the nanochannel is larger.     

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。     

Mechanisms behind concentration profiles illustrated by charge and concentration distributions around ions in double layers

The mechanisms behind the behaviour of concentration and charge density profiles in diffuse electric double layers are investigated quantitatively for 1:1 and 2:2 electrolytes. This is done by analysing various contributions to the mean force that acts on each ion. The forces are obtained from the calculated ionic charge and concentration distributions around individual ions at various positions in the double layer. These distributions are presented graphically which allows an immediate visual illustration of the mechanisms in action. Some features studied are charge inversion in double layers for divalent aqueous electrolytes, overcompensation of surface charge due to large amounts of physisorbed counterions, ion size effects in the double layer structure and various mechanisms that cause deviations from the predictions of the Poisson–Boltzmann approximation. A major objective of the paper is to present the results in a visual form and explain aspects of modern double layer theory in a simple manner.     

Influence of monovalent ion size on colloidal forces probed by Monte Carlo simulations

The present work studies the role of ionic size in the interactions between the electrical double layers of colloids immersed into electrolyte solutions of monovalent ions. Such interactions are studied by means of Monte Carlo (MC) simulations and the classical Derjaguin-Landau-Verwey-Overbeek (DLVO) theory. Despite the omission of the steric effects and some other features of real electrolyte solutions, DLVO theory is known to work qualitatively well for 1:1 electrolyte solutions. However, this affirmation is based on previous tests where an ionic diameter around 0.4 nm was taken for all ionic species. In contrast, some experimental studies suggest that larger hydrated ions should be considered and even specified for each type of ion. In this work, the importance of ionic size is analyzed by applying the primitive model of electrolyte to the intermediate region between a pair of equally charged infinite planar surfaces. The double layer interactions were calculated from the ionic densities at the distance of closest approach to the charged surfaces, this method constitutes an alternative to the traditional calculations at the midplane. Our MC simulations predict the existence of negative net pressures for monovalent electrolytes in the case of zero charge density. In addition, MC simulations reveal some disagreements with theoretical predictions for ionic diameters larger than 0.4 nm. These discrepancies can become significant if surface charge density is large enough due to the restructuration of the double layer. The physical mechanisms for these deviations are also discussed.     

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.     

Understanding anomalous current-voltage characteristics in microchannel-nanochannel interconnect devices

The integration of a microchannel with a nanochannel is known to exhibit anomalous nonlinear current-voltage characteristics. In this paper, we perform detailed numerical simulations considering a 2-D nonlinear ion transport model, to capture and explain the underlying physics behind the limiting resistance and the overlimiting current regions, observed predominantly in a highly ion-selective nanochannel. We attribute the overlimiting current characteristics to the redistribution of the space charges resulting in an anomalous enhancement in the ionic concentration of the electrolyte in the induced space charge region, beyond a critical voltage. The overlimiting current with constant conductivity is predicted even without considering the effects of fluidic nonlinearities. We extend our study and report anomalous rectification effects, resulting in an enhancement of current in the non-ohmic region, under the application of combined AC and DC electric fields. The necessary criteria to observe these enhancements and some useful scaling relations are discussed.     

Nanofluidic channels by anodic bonding of amorphous silicon to glass to study ion-accumulation and ion-depletion effect

A unique phenomenon, ion-enrichment and ion-depletion effect, exists in nanofluidic channels and is observed in amorphous silicon (α-Si) nanochannels as shallow as 50 nm. As a voltage is applied across a nanochannel, ions are rapidly enriched at one end and depleted at the other end of the nanochannel. α-Si is deposited on glass by plasma enhanced chemical vapor deposition and is selectively etched to form nanochannels. The depth of nanochannels is defined by the thickness of the α-Si layer. Low temperature anodic bonding of α-Si to glass was used to seal the channel with a second glass wafer. The strength of the anodic bond was optimized by the introduction of a silicon nitride adhesion promoting layer and double-sided bonding resulting from the electric field reversal. Completed channels, 50 nm in depth, 5 micron wide, and 1 mm long were completely and reliably sealed. Structures based on nanochannels 50-300 nm deep were successfully incorporated into nanofluidic devices to investigate ionic accumulation and depletion effect due to overlapping of electric double layer.     

Molecular dynamics simulations of ion transport through carbon nanotubes. III. Influence of the nanotube radius, solute concentration, and applied electric fields on the transport properties

The present investigations continue previous research on transport in aqueous ionic solutions through carbon nanotubes. Specifically, the effects of the nanotube radius, solute concentration, and applied external electric fields on the transport properties are investigated in terms of mobilities, currents, and pairing times of the solute ions. The simulated transport features are corroborated with general theoretical results of nanofluidics (such as the linear log-log regime of the nanochannel conductance as function of the solute concentration and the current-voltage curve of the channel). Discontinuities in the partial ionic currents are explained on the basis of a recent theoretical model of quantized ionic conductance in nanopores, developed by Zwolak et al. Correlations between the structural and dynamic properties are established, linking causally the highly structured spatial density profiles, the ion pairing phenomenon and the ionic currents.     

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.     

Ion separation in nanofluidics

Ionic species with a constant charge-to-size ratio (i.e. electrophoretic mobility) cannot be separated in electroosmotic or pressure-driven flow along microscale channels. In nanoscale channels, however, the enormous electric fields inside electrical double layers cause transverse ion distributions yielding charge-dependent mean ion speeds in the flow. Those ions with a constant charge-to-size ratio can thus be separated solely by charge (or equivalently, size) in nanofluidics. Here we develop an analytical model to optimize and compare the separation of such ions in nanochannel chromatography and nanochannel electrophoresis in terms of selectivity, plate height and resolution. Both planar and cylindrical geometries are considered. It is found that in nanoscale channels chromatography yields a larger selectivity and a larger minimum reduced plate height than electrophoresis does. The maximum resolution is, however, comparable between these two nanofluidic approaches, where the optimal channel half-height or tube radius is within the range of 1-10 times the Debye length. Our results also suggest that cations can be better separated in nanofluidics than can anions.     

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.     

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.     

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.     

Large scale lithography-free nano channel array on polystyrene

This paper reports a new fabrication method of lithography-free nanochannel array. It is based on the cracking process on the surface of a polystyrene (PS) Petri-dish, one type of thermoplastic that is composed of uni-axial macromolecular chains. Under proper conditions, parallel nanochannels with equal interspaces are obtained. Control over the channel depth from 20 nm to 200 nm is achieved, with the channel length reaching tens of millimetres. The PDMS replication based on PS nanochannel array has been successfully carried out. In combination with the microstructure, both an ion enrichment device and a current rectification device are fabricated, and their quantified characters manifested the applicability of the channel array structure in nanofluidics.     

An electrodynamics-based model for ion diffusion in microbial polysaccharides

An electrodynamics-based model was formulated for simulation of ion diffusion in microbial polysaccharides. The fixed charges and electrostatic double layers that may associate with microbial polysaccharides and their effects on ion diffusion were explicitly built into the model. The model extends a common multicomponent ion diffusion formulation that is based on irreversible thermodynamics under a zero ionic charge flux condition, which is only applicable to the regions without fixed charges and electrostatic double layers. An efficient numerical procedure was presented to solve the differential equations in the model. The model well described key features of experimental observations of ion diffusion in negatively charged microbial polysaccharides including accelerated diffusive transport of cations, exclusion of anions, and increased rate of cation transport with increasing negative charge density. The simulated diffusive fluxes of cations and anions were consistent with a cation exchange diffusion concept in negatively charged polysaccharides at the interface of plant roots and soils; and the developed model allows to mathematically study such diffusion phenomena. An illustrative example was also provided to simulate dynamic behavior of ionic current during ion diffusion within a charged bacterial cell wall polysaccharide and the effects of the ionic current on the compression or expansion of the bacterial electrostatic double layer at the interface of the cell wall and bulk solution.     

The sedimentation potential in a dilute suspension

During the sedimentation of charged particles immersed in an ionic solution, a gradient of electric potential forms. The gradient reflects the surface potential of the particles, moderated by the screening effect of the diffuse charge. In the theory developed here the fact that the net current is zero is used to derive an expression for the gradient which is free of ambiguities associated with the shape of the averaging volume, one of the shortcomings of the previous theory. The general expression also differs from those derived by previous investigators who summed the dipole fields for each particle, a procedure which leads to expressions that fail to satisfy the constraint on the current. The effect of the macroscopic field on the sedimentation velocity is found to be significant when the double layer is thick. Changes in the sedimentation coefficient are also reflected in the Brownian diffusivity of suspended particles but here the effect appears less significant.