Analytical treatment of electrokinetic microfluidics in hydrophobic microchannels

Microfluidics in microelectromechanical systems (MEMS) and nanoelectromechanical systems (NEMS) devices is complex due to the large surface area to volume ratio. Thus, surface properties play an important role in flow behavior. In this paper, we summarize the effects of electric double layer and surface hydrophobicity of rectangular microchannels on time-dependent electrokinetic flow. Theoretically, we have shown that flow resistance can, in principal, be significantly reduced so that a larger flow rate can be obtained for pressure-driven flow or electric-field-driven flow. This relies on the ability to change surface charges and surface hydrophobicity independently. Our theoretical results provide guidelines for the design and operation of microfluidic flow in rectangular microchannels. Because of liquid slippage, zeta potential determination by traditional method could be overestimated. Taking into account the effect of hydrophobicity, a modified method is proposed to determine the zeta potential and slip coefficient for parallel-plate microchannels with hydrophobic surfaces.     

On the surface conductance, flow rate, and current continuities of microfluidics with nonuniform surface potentials

The characteristics of electrokinetic flow in a microchannel depend on both the nature of surface potentials, that is, whether it is uniform or nonuniform, and the electrical potential distribution along the channel. In this paper, the nonlinear Poisson-Boltzmann equation is used to model the electrical double layer and the lattice Boltzmann model coupled with the constraint of current continuity is used to simulate the microfluidic flow field in a rectangular microchannel with a step variation of surface potentials. This current continuity, including surface conduction, convection, and bulk conduction currents, has often been neglected in the literature for electroosmotic flow with nonuniform (heterogeneous) microchannels. Results show that step variation of ion distribution caused by step variation surface potential will influence significantly the electrical potential distribution along the channel and volumetric flow rate. For the system considered, we showed that the volumetric flow rate could have been overestimated by as much as 70% without consideration of the current continuity constraint.     

Effect of skimming layer in an electroosmotically driven viscoelastic fluid flow over charge modulated walls

We report a numerical study on the effect of the skimming layer in an EOF of Oldroyd-B fluid over charge modulated walls. Three types of flow conditions were identified on the basis of the relative thickness of the skimming layer and the electrical double layer. We observe maximum slip velocity magnitude when the skimming layer thickness is very less than the thickness of the electrical double layer. For higher skimming layer thickness compared to the thickness of electrical double layer, slip velocity magnitude attenuates, and the polymeric stress inside the skimming layer becomes zero. Enhanced fluid elasticity generates asymmetric flow structures inside the microchannel, which can also be achieved by imposing an asymmetric surface charge along the channel walls. Our present analysis highlights the complex flow dynamics of the EOF of biofluids/polymeric fluids with a near-wall region depleted of macro-molecules.     

Electroosmotic transport of immiscible binary system with a layer of non‐conducting fluid under interfacial slip: The role applied pressure gradient

We investigate the transport of immiscible binary fluid layers, constituted by one conducting (top layer fluid) and another non‐conducting (bottom layer fluid) fluids in a microfluidic channel under the combined influences of an applied pressure gradient and imposed electric field. We solve the transport equation governing the flow dynamics analytically and obtain the closed‐form expressions of the velocity fields. We bring out the alteration in the flow dynamics, mainly attributable to the non‐linear interaction between interfacial slip and the electrical double layer effect over small scales as modulated by the applied pressure gradient. In particular, we show the augmentation in the net volume transport rate through the channel, emerging from an intricate competition among electrical forcing, applied pressure gradient and the viscous resistance as modulated by the interfacial slip. We believe that the results of this study may be of immense consequence for the design of various microfluidic devises, which are often used for the manipulation of two immiscible fluids in different biomedical/biochemical processes.     

Starting electroosmotic flow in an annulus and in a rectangular channel

The initial electroosmotic flow through a small pore or microchannel with annular or rectangular cross section is studied under the Debye-Hückel approximation. Analytical series solutions and their asymptotic behavior for small and large non-dimensional electrokinetic widths are found for these two basic cases. The explicit and accurate solutions are particularly useful for examining various geometric/physical effects on the transient time scales and the flow rates for the transient states. The steady flow rate of the smaller channel may be disproportionately smaller than a reference channel if the electric double layer is thick, but will be in close proportion to the area ratio if the electric double layer is thin. A smaller channel compared to a reference channel has a shorter transient time scale, and the transient flow has characters very different from the steady state if the electric double layer is thin. The total transient flow rate of several smaller pores or channels may exceed largely that of a single large pore or channel with the same total cross section on the transient time scale of the smaller channels. The results have important implications on liquid transport in micropores or channels by pulse voltages or more general time-varying voltages.     

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.     

Improved understanding of the effect of electrical double layer on pressure-driven flow in microchannels

Traditionally, the effects of electrical double layer on pressure-driven flow in microchannels were modeled by using the Poisson-Boltzmann equation and the fluid momentum equation with a flow-induced body force term. Such a model, however, usually underestimate the electrical double layer effects on the flow. In this study, a theoretical model of the electrical double layer field is developed to provide a better understanding of the electrical double layer effects. The electrical potential and ionic concentration distribution in dilute solutions in small microchannels are investigated by numerically solving this new model. This newly developed model predicted the deficit of counter-ions in the bulk liquid region due to the accumulation of counter-ions in the EDL region, and the surplus of co-ions in the bulk liquid region due to rejection of the co-ions in the EDL region. The presence of the net charges in the bulk liquid region is responsible for the strong electroviscous effects in dilute solutions in small microchannels.     

Electro-Viscous Effects on Liquid Flow in Microchannels

The presence of the electrical double layer near a solid-liquid interface results in the electro-viscous effect on pressure-driven liquid flow through microchannels. The objective of this paper is to examine the magnitude of the additional flow resistance caused by the electrokinetic effect in microchannels. Deionized ultrafiltered water, 10(-4) and 10(-2) M aqueous KCl solutions, 10(-4) M AlCl(3) solution, and 10(-4) M LiCl solution were used as the testing liquids. Carefully designed flow measurements were conducted in three silicon microchannels with a height of 14.1, 28.2, and 40.5 μm, respectively. The measured dP/dx for the pure water, the 10(-4) M KCl solution, and the 10(-4) M LiCl solution was found to be significantly higher than the prediction of the conventional laminar flow theory at the same Reynolds number. Such a high flow resistance and the resulting high apparent viscosity strongly depend on the channel's height, the ionic valence, and the concentration of the liquids. The zeta potentials for the liquid-solid systems were calculated by using the measured streaming potential data. The experimentally determined dP/dx approximately Re relationships were compared with the predictions of a theoretical electro-viscous flow model, and a good agreement was found for pure water, 10(-4) M KCl solution, and 10(-4) MAlCl(3) solution systems. The present electrokinetic flow model cannot interpret the flow characteristics of the LiCl solution. Copyright 2001 Academic Press.     

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.     

Electroosmotic flow: From microfluidics to nanofluidics

Electroosmotic flow (EOF), a consequence of an imposed electric field onto an electrolyte solution in the tangential direction of a charged surface, has emerged as an important phenomenon in electrokinetic transport at the micro/nanoscale. Because of their ability to efficiently pump liquids in miniaturized systems without incorporating any mechanical parts, electroosmotic methods for fluid pumping have been adopted in versatile applications—from biotechnology to environmental science. To understand the electrokinetic pumping mechanism, it is crucial to identify the role of an ionically polarized layer, the so-called electrical double layer (EDL), which forms in the vicinity of a charged solid–liquid interface, as well as the characteristic length scale of the conducting media. Therefore, in this tutorial review, we summarize the development of electrical double layer models from a historical point of view to elucidate the interplay and configuration of water molecules and ions in the vicinity of a solid–liquid interface. Moreover, we discuss the physicochemical phenomena owing to the interaction of electrical double layer when the characteristic length of the conducting media is decreased from the microscale to the nanoscale. Finally, we highlight the pioneering studies and the most recent works on electro osmotic flow devoted to both theoretical and experimental aspects.     

Simultaneous estimation of zeta potential and slip coefficient in hydrophobic microchannels

Electroosmotic flows through hydrophobic microchannels experience velocity slip at the channel wall, which increases the volumetric flow rate at a given electric potential gradient. The conventional method of zeta potential estimation using the volumetric flow rate may yield quite inaccurate zeta potential unless the velocity slip is appropriately taken care of. In the present investigation we develop a method for simultaneous estimation of zeta potential and velocity slip coefficient in the electroosmotic flow through a hydrophobic microchannel using velocity measurements. The relevant inverse problem is solved through the minimization of a performance function utilizing a conjugate gradient method. The present method is found to estimate the zeta potential and slip coefficient accurately even with noisy velocity measurements.     

A method for simultaneous estimation of inhomogeneous zeta potential and slip coefficient in microchannels

In the microchannels made of hydrophobic materials, the fluid velocity is determined by the zeta potential and velocity slip, both of which may be inhomogeneous due to the adsorption of protein to the channel wall. The inhomogeneity of zeta potential and slip coefficient sometimes causes recirculating flows which in turn affect the transport and mixing of solutes through the microchannels. In the present investigation we devise a method for the simultaneous estimation of inhomogeneous zeta potential and inhomogeneous slip coefficient using velocity measurements. A conjugate gradient method supplemented by the adjoint variable method is adopted in the solution of the relevant inverse problem to reduce the computational burden. The present method is found to estimate the inhomogeneous zeta potential and the slip coefficient simultaneously even with noisy velocity measurements. This method is expected to contribute to the optimal design and robust operation of various microfluidic devices, where the flow patterns and the volumetric flow rates are critically influenced by the profiles of inhomogeneous zeta potential and inhomogeneous slip coefficient.     

Theory of transport in nanofluidic channels with moderately thin electrical double layers: effect of the wall potential modulation on solutions of symmetric and asymmetric electrolytes

Electrokinetic phenomena play an important role for the transport in submicrometer-size channels since the electric double layers formed at the walls can occupy a substantial part of the channel volume. This presents a theoretical difficulty and specific problems are usually treated numerically or not comprehensively. In our work we present a theoretical model that allows one to obtain analytical expressions for the transport of fluid (electro-osmotic flow), ions (electric current), and dissolved charged molecules (analytes). The model is based on the weak double layer approximation and has a wide range of validity. An important feature of this theoretical approach is that it is applicable not only to symmetric but also to asymmetric 2:1 and 1:2 electrolytes which exhibit very interesting properties in nanoscale channels. The possibility of affecting the wall electrokinetic zeta potential by applying a transverse voltage bias is analyzed. This transverse bias is used in an attempt to control the transport in the channel and such devices are often called "fluidic field-effect transistors." Our model quantifies the effect of the voltage bias on the zeta potential of the channel wall and therefore can be used for prediction of transport and optimization of separations in such fluidic devices.     

ac electroosmosis in rectangular microchannels

Motivated by the growing interest in ac electroosmosis as a reliable no moving parts strategy to control fluid motion in microfluidic devices for biomedical applications, such as lab-on-a-chip, we study transient and steady-state electrokinetic phenomena (electroosmosis and streaming currents) in infinitely extended rectangular charged microchannels. With the aid of Fourier series and Laplace transforms we provide a general formal solution of the problem, which is used to study the time-dependent response to sudden ac applied voltage differences in case of finite electric double layer. The Debye-Huckel approximation has been adopted to allow for an algebraic solution of the Poisson-Boltzmann problem in Fourier space. We obtain the expressions of flow velocity profiles, flow rates, streaming currents, as well as expressions of the complex hydraulic and electrokinetic conductances. We analyze in detail the dependence of the electrokinetic conductance on the extension of linear dimensions relative to the Debye length, with an eye on finite electric double layer effects.     

Nonequilibrium electrokinetic effects in beds of ion-permselective particles

Electrokinetic transport of fluorescent tracer molecules in a bed of porous glass beads was investigated by confocal laser scanning microscopy. Refractive index matching between beads and the saturating fluid enabled a quantitative analysis of intraparticle and extraparticle fluid-side concentration profiles. Kinetic data were acquired for the uptake and release of electroneutral and counterionic tracer under devised conditions with respect to constant pressure-driven flow through the device and the effect of superimposed electrical fields. Transport of neutral tracer is controlled by intraparticle mass transfer resistance which can be strongly reduced by electroosmotic flow, while steady-state distributions and bead-averaged concentrations are unaffected by the externally applied fields. Electrolytes of low ionic strength caused the transport through the charged (mesoporous) beads to become highly ion-permselective, and concentration polarization is induced in the bulk solution due to the superimposed fields. The depleted concentration polarization zone comprises extraparticle fluid-side mass transfer resistance. Ionic concentrations in this diffusion boundary layer decrease at increasing field strength, and the flux densities approach an upper limit. Meanwhile, intraparticle transport of counterions by electromigration and electroosmosis continues to increase and finally exceeds the transport from bulk solution into the beads. A nonequilibrium electrical double layer is induced which consists of mobile and immobile space charge regions in the extraparticle bulk solution and inside a bead, respectively. These electrical field-induced space charges form the basis for nonequilibrium electrokinetic phenomena. Caused by the underlying transport discrimination (intraparticle electrokinetic vs extraparticle boundary-layer mass transfer), the dynamic adsorption capacity for counterions can be drastically reduced. Further, the extraparticle mobile space charge region leads to nonlinear electroosmosis. Flow patterns can become highly chaotic, and electrokinetic instability mixing is shown to increase lateral dispersion. Under these conditions, the overall axial dispersion of counterionic tracer can be reduced by more than 2 orders of magnitude, as demonstrated by pulse injections.     

Electrokinetics over hydrophobic surfaces

Segregation of phases in the vicinity of hydrophobic surfaces may turn out to be immensely consequential towards altering the coupling of electrostatics and hydrodynamics over interfacial scales. Here, we review the fundamental advances towards bringing out the various facets of electrokinetic transport over hydrophobic interfaces. We lay significant emphasis on the developments in understanding the slippery electrohydrodynamics over such interfaces, by appealing to the considerations across various spatio‐temporal scales as unveiled by molecular dynamics as well as mesoscopic modelling paradigms (such as phase field and lattice Boltzmann). We envisage that despite significant advancements being achieved towards relating the macroscopic slip‐length with the underlying molecular or mesoscopic phenomena, future efforts could be directed towards developing more robust statistically based models that may connect rarefied gas dynamics in the segregated phase with bulk electrokinetic transport and possible giant augmentations in the consequent fluid flow.     

Applications of electrokinetic phenomena in materials science

The discovery of electrokinetic phenomena by Reuss in 1808 and further investigations that gave rise to the concept of the electrical double layer have played an important role in the understanding of colloidal stability. Electrokinetic phenomena are a family of effects in which a liquid moves tangentially to a charged surface. Well-known phenomena of this kind are electrophoresis, electro-osmosis, streaming potential, and sedimentation potential. A review of parameters involved in the electrochemistry of suspensions is made. The practical applications of these phenomena have become widespread in a broad range of research fields such as biomaterials, biofilms, electrokinetic waste remediation, membranes, nuclear and fossil-fired power plants, adhesive and sealant science, and concrete science. The purpose of this paper is to provide an overview of electrokinetic phenomena and their application to surface modification and characterization in a large number of research fields such as corrosion and protection processes, environmental remediation (soil and sediments, transport processes, inorganic pollutants, solid particle surfaces, filter membranes, and biosorption phenomena), cement-based systems, and biological systems.     

Determination of the Navier slip coefficient of microchannels exploiting the streaming potential

For most microchannels made of hydrophobic materials such as polymers, velocity slip occurs at the wall, affecting volumetric flow rate of electroosmotic flow Q(eof) and streaming potential (??(str)/?z). Since most techniques exploit Q(eof) or (??(str)/?z) to determine the zeta potential, ζ, it is very difficult to measure ζ of hydrophobic walls, if the slip coefficient b is not found a priori. Until now, Q(eof) and (??(str)/?z) are known to depend on ζ and b in a same functional form, which makes it impossible to estimate ζ or b separately using measurements of Q(eof) and (??(str)/?z). However, exploiting the analytic formula for Q(eof) and (??(str)/?z) derived in the present work, it is found that the effect of ζ and that of b on Q(eof) and (??(str)/?z) can be separated from each other by varying the bulk ionic concentration. Thus, the slip coefficient as well as the zeta potential of hydrophobic microchannels can be found with reasonable accuracy by means of a nonlinear curve fitting method using measured data of Q(eof) and (??(str)/?z) at various bulk ionic concentrations. The present method allows an accurate estimation of slip coefficient of hydrophobic microchannels, which is quite simple and cheap compared with methods employing microparticle velocimetry.     

Recent advances in direct current electrokinetic manipulation of particles for microfluidic applications

Microfluidic devices have been extensively used to achieve precise transport and placement of a variety of particles for numerous applications. A range of force fields have thus far been demonstrated to control the motion of particles in microchannels. Among them, electric field‐driven particle manipulation may be the most popular and versatile technique because of its general applicability and adaptability as well as the ease of operation and integration into lab‐on‐a‐chip systems. This article is aimed to review the recent advances in direct current (DC) (and as well DC‐biased alternating current) electrokinetic manipulation of particles for microfluidic applications. The electric voltages are applied through electrodes that are positioned into the distant channel‐end reservoirs for a concurrent transport of the suspending fluid and manipulation of the suspended particles. The focus of this review is upon the cross‐stream nonlinear electrokinetic motions of particles in the linear electroosmotic flow of fluids, which enable the diverse control of particle transport in microchannels via the wall‐induced electrical lift and/or the insulating structure‐induced dielectrophoretic force.