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Hypothesis We investigate if the shear-stress exerted on the wall of a glass microchannel can be a robust and accurate criterion for the safe electro-osmotic transfer of polyethylene oxide (PEO) chains dissolved in a NaCl aquatic solvent. To this end, a comprehensive multiscale formulation based on the rheological and electrochemical modeling of the PEO dynamics is proposed. Phenomena that occur in microscale, e.g., the migration of PEO to the core region of the channel and Polymeric Depletion Layer (PDL) formation, and in nanoscale, e.g., the development of an electric double layer on the glass surface and ionic steric effects, are included. Experimental arrangement We study the electro-osmotic flow of PEO solutions (0.1–0.5%), flowing in a glass microchannel of rectangle shape, with dimensions of 300 μm in length and 75 μm in height. We vary the externally applied electric field (300–500 V/cm), and the bulk ionic concentration (0.001–10 mM). Findings We find that all features of our formulation are indeed essential to reproduce the experimental data of Huang, Chen, Wong, Liow, Soft Matter, (2016) precisely. Although the PDL formation preserves the fragile nature of biopolymers, the dominant stress is the normal stress, and the critical value is at the PDL interface. A new design criterion for microdevices is proposed.

Electrokinetic effects in porous media play a key role in a number of natural and industrial processes. Applications such as enhanced oil recovery, soil remediation and even drug delivery are affected by the Coulombic forces created by the solid-fluid interfacial interactions. These electrokinetic effects promote the development of non-homogenous slipping flow over charged surfaces at the pore scale, which can have a significant impact in the hydrodynamics of tight porous materials. For transport of ionic solutions in such systems (e.g. transport of low salinity water in tight oil reservoirs), combined effect of hydrodynamic transport and electrokinetic transport would be expected. While transport in pressure-driven transport will be pronounced in high permeability flow pathways, transport due to electric fields (e.g. electro-osmosis) will be more pronounced in tight pores were electrical diffuse layer is not negligible. In this work, we explored the pore-scale hydrodynamic characteristics of charged porous media using computational fluid dynamics. Different flow driving mechanisms were studied, e.g. conventional pressure driven flow, pure electro-osmosis as well as their superposition under different amounts of charged material. We then analyzed the effect of these distinct flow regimes on the transport of a passive tracer, finding how different driving mechanism result in distinct dispersion and mixing characteristics.

Surface charge at solid-electrolyte interface is generally coupled with the local electrolyte properties (ionic concentration, pH, etc.), and therefore not as assumed homogeneous on the solid surfaces in the previous studies. The inhomogeneous charge brings huge challenges in predictions of electro-osmotic transport and has never been well studied. In this work, we first propose a classification of electro-osmosis based on a dimensionless number which is the ratio of the Debye length to the characteristic pore size. In the limit of thin electrical double layer, we establish a pore-scale numerical model for inhomogeneously charged electro-osmosis including four ions: Na+,Cl-,H+ and OH-. Based on reconstructed porous media, we simulate the electro-osmosis with inhomogeneous charge using lattice Boltzmann method. The nonlinear response of electro-osmotic velocity to applied electrical field and the reverse flow have been observed and analyzed.

We generate and study electroosmotic shear flow in microchannels. By chemically or electrically modifying the surface potential of the channel walls a shear flow component with controllable velocity gradient can be added to the electroosmotic flow caused by double layer effects at the channel walls. Chemical modification is obtained by treating the channel wall with a cationic polymer. In case of electric modification, we used gate electrodes embedded in the channel wall. By applying a voltage to the gate electrode, the zeta potential can be varied and a controllable, uniform shear stress can be applied to the liquid in the channel. The strength of the shear stress depends on both the gate voltage and the applied field which drives the electroosmotic shear flow. Although the stress range is still limited, such a microchannel device can be used in principle as an in situ micro-rheometer for lab on a chip purposes.

In this paper, we conducted numerical simulation of the electroosmotic flow in a column of an aqueous solution surrounded by an immiscible liquid. While governing equations in this case are the same as that in the electroosmotic flow through a microchannel with solid walls, the main difference is the types of interfacial boundary conditions. The effects of electric double layer (EDL) and surface charge (SC) are considered to apply the most realistic model for the velocity boundary condition at the interface of the two fluids. Effects on the flow field of ς-potential and viscosity ratio of the two fluids were investigated. Similar to the electroosmotic flow in microchannels, an approximately flat velocity profile exists in the aqueous solution. In the immiscible fluid phase, the velocity decreases to zero from the interface toward the immiscible fluid phase. The velocity in both phases increases with ς-potential at the interface of the two fluids. The higher values of ς-potential also increase the slip velocity at the interface of the two fluids. For the same applied electric field and the same ς-potential at the interface of the two fluids, the more viscous immiscible fluid, the slower the system moves. The viscosity of the immiscible fluid phase also affects the flatness of the velocity profile in the aqueous solution.

A mathematical framework for analysing electrokinetic flow in microchannel networks is outlined. The model is based on conservation of volume and total charge at network junctions, but in contrast to earlier theories also incorporates conservation of ion charge there. The model is applied to mixed pressure-driven/electro-osmotic flows of binary electrolytes through homogeneous microchannels as well as a 4:1:4 contraction-expansion series network. Under conditions of specified volumetric flow rate and ion currents, non-linear steady-state phenomena may arise: when the direction of the net co-ion flux is opposite to the direction of the net volumetric flow, two different fully developed, steady-state flow solutions may be obtained. Model predictions are compared with two-dimensional computational fluid dynamics (CFD) simulations. For systems where two steady states are realisable, the ultimate steady behaviour is shown to depend in part upon the initial state of the system.

Existing microfluidic circuit theories consider conservation of volume and conservation of total charge at each channel intersection (node) that exists within a circuit. However, in a strict sense conservation of number (or charge) for each ion species that is present should also be applied. To be able to perform such a conservation the currents due to the movement of each ion species (electrokinetic ion currents) that occur within each channel need to be known. Hence, we here present analytical and numerical methods for calculating these ion currents (and fluid flowrates) in Newtonian binary electrolyte solutions flowing within two-dimensional thin slits and pipes. Analytical results are derived in the limits of low potential, high potential, and thin double layers. We show that irrespective of double layer overlap, the Boltzmann distribution is valid provided that a local geometric mean is used for the reference ion concentration. While the real significance of the work lies in its application to multi-channel microfluidic circuit theory (see the accompanying paper of Biscombe et al. [1]), the present results show that even in single channels, ion current behaviour can be surprisingly complex.

A two-fluid electroosmotic flow in a microchannel is studied by considering full hydrodynamic and electrostatic interactions on the interface. Jumps in electrical potential and in charge density across the interface, in particular, are found to create counterintuitive flow behavior through the electrostatic interaction of the interface with the external field imposed. The interfacial electrostatic effects are shown to induce flow reversal within physically reasonable parametric ranges. It is also shown that the electrostatic properties of the interface must be carefully considered in electroosmotic pumping lest the nonconducting fluid should stay stationary or flow in an unintended direction. A formula for quantitative control of electroosmotic pumping is provided.

Optical tweezers enable one to trap a single particle without any mechanical contact and to measure its position and the forces acting on it with high resolution (±4 nm, ±160 fN). Taking advantage of a specially designed microfluidic cell the electrophoretic response of the colloid under study and the electroosmotic effect on the surrounding medium are determined using the identical colloid. The former is found to be by more than one order of magnitude larger than the electroosmotic effect. It is shifted in phase with respect to the external field, hence giving rise to a complex electrophoretic mobility which can be theoretically described by a strongly damped driven harmonic oscillator model. By exchanging the medium surrounding the colloid it is possible to deduce the (KCl) concentration dependence of the single colloid electrophoretic response. The results are compared with conventional Zetasizer measurements.

Electrophoresis of a spherical composite particle normal to a plane is investigated theoretically. The composite particle under consideration, or the "soft" particle, consists of an inner hard core coated with a concentric porous layer containing uniformly distributed fixed charges. A pseudo-spectral method based on a Chebyshev polynomial is adopted to solve the resulting electrokinetic equations. The effects of general parameters of electrokinetic interest are examined, such as the double layer thickness, the density of the fixed charges carried in the porous layer, the retarding friction coefficient of the porous layer, and so forth. In particular, the effect of the presence of the planar boundary is examined in detail, including the clearly visible deformation of the double layer due to the presence of the boundary and its electrokinetic implications. Local maximum of charge distribution within the porous layer near the north pole is observed when the double layer is thick. Moreover, the impact of the electroosmotic flow within the porous layer is explored. The appearance of a vortex flow due to electroosmosis within the porous layer is shown, which explains directly the retarding effect of mobility when the double layer becomes thinner, as the orientation of this vortex is opposite to the particle motion. Comparisons with various limiting cases available in the literature show excellent agreement, indicating the accuracy and reliability of the results presented in this study.

The vulnerability of vesicles to electroporation and rupture by externally applied electric fields, combined with the ability of dielectrophoresis and/or AC electroosmosis to manipulate suspended vesicles over micropatterned electrodes suggests new techniques to electrically trigger localized chemical reactions at predetermined positions in microfluidic devices. The electric field conditions needed to rupture giant unilamellar phospholipid vesicles were determined as a function of vesicle size in a simple coplanar microelectrode array geometry. Rupture results were interpreted in terms of the spatially varying electric field strength, calculated via the Poisson equation and accounting for frequency effects on electrode impedance, and the experimentally measured vesicle elevation. The vesicle transmembrane voltage scales linearly with the applied electric field strength according to the Schwan theory of electroporation, so that larger vesicles are usually more prone to electric field induced rupture than smaller ones in the uniform electric fields that are typically employed to cause electroporation and rupture. Yet, in the coplanar microelectrode arrangement, larger vesicles preferentially reside at larger elevations where the local field strengths are weaker. As a result, there is a sensitive range of vesicle radii that are most prone to electric field induced rupture over a micropatterned electrode array that leaves the largest vesicles resistant to rupture.

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–Huckel 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.

Spatially two-dimensional nonequilibrium mathematical model describing electroosmotic flow through a submicrometer channel with an electric charge fixed on the channel walls is presented. This system is governed by the hydrodynamic, electrostatic, and mass transport phenomena. The model is based on the coupled mass balances, Poisson, Navier-Stokes, and Nernst-Planck equations. Nonslip boundary conditions are employed. The effect of an imposed electric field on the system behavior is studied by means of a numerical analysis of the model equations. We have obtained the following findings. If the channel width is comparable to the thickness of the electric double layer, the system behaves as an ion-exchange membrane and the dependence of the electric current passing through the channel on the applied voltage is strongly nonlinear. In the case of negatively (positively) charged walls, a narrow region of very low conductivity (so-called ionic gate) is formed in the free electrolyte near the channel entry facing the anode (cathode) side. For a wide channel, the electric current is proportional to the applied voltage and the velocity of electrokinetic flow is linearly proportional to the electric field strength. Complex hydrodynamics (eddy formation and existence of ionic gates) is the most interesting characteristics of the studied system. Hence, current-voltage and velocity-voltage curves and the corresponding spatial distributions of the model variables at selected points are studied and described in detail.

Approximate analytical formulae valid for any porous media with elongated pores are derived for the electro-osmotic coefficient alpha and for the average ionic concentration n . A macroscopic Debye-Hückel length kappa (-)(-1) based on n is introduced. Simultaneously, the electro-osmotic coefficient alpha is systematically calculated for various media, zeta potentials and electrolyte concentrations by solving the local equations. Numerical results show that kappa (-)(-1) and alpha follow universal curves valid whatever the porous medium; these curves can be approximated by the analytical formulae previously derived. These formulae can be used to provide a priori estimates of the electro-osmotic coefficient.

Electroosmosis in homogeneously charged micro- and nanoscale random porous media has been numerically investigated using mesoscopic simulation methods which involve a random generation-growth method for reproducing three-dimensional random microstructures of porous media and a high-efficiency lattice Poisson-Boltzmann algorithm for solving the strongly nonlinear governing equations of electroosmosis in three-dimensional porous media. The numerical modeling and predictions of EOF in micro- and nanoscale random porous media indicate that the electroosmotic permeability increases monotonically with the porosity of porous media and the increasing rate rises with the porosity as well; the electroosmotic permeability increases with the average solid particle size for a given porosity and with the bulk ionic concentration also; the proportionally linear relationship between the electroosmotic permeability and the zeta potential on solid surfaces breaks down for high zeta potentials. The present predictions agree well with the available experimental data while some results deviate from the predictions based on the macroscopic theories.

Both from the experimental and theoretical viewpoints it is of fundamental importance to know precisely which are the fluid flow characteristics in a (cylindrical, say) closed cell under the action of an externally applied electric field, parallel to the cell axis. This is so because in many cases the experimental determination of the electrophoretic mobility of dispersed particles is carried out in closed cells, whereby the motion of the particles in the laboratory reference system is the result of the superposition of their electrophoretic migration plus the liquid motion with respect to the cell. This makes it of utmost importance to analyze the above-mentioned fluid and particle movements. If, in particular, this evaluation is carried out in the presence of alternating fields of different frequencies, information about the dynamics and time scales of the processes involved can be obtained for different frequencies of the applied field. In the present contribution, we discuss experimental results based on the determination of the velocity of polystyrene latex particles in a closed, cylindrical electrophoresis cell, and compare them to our previous theoretical analysis of the problem. It is concluded that the theory explains with great accuracy the observed particle velocities. In addition to the use of the particles as probes for the fluid velocity distribution, this work intends to give additional clues on the frequencies and positions for which electrophoretic mobility measurements in closed cells can be more reliable.

The present study deals with thermal transport characteristics of an electrolyte solution flowing through a slit nanochannel with polyelectrolyte walls, known as soft nanochannel. The sources of the fluid flow are the pressure gradient along the channel axis and the electrokinetic effects that trigger an electroosmotic flow under the impact of a uniformly applied electric field. The polyelectrolyte layer (PEL) is denoted as a fixed charge layer (FCL) and the electrolyte ions can be present both inside and outside the PEL. Therefore, the PEL-electrolyte interface acts as a semi-penetrable membrane. To the best of our knowledge, the thermal analysis of mixed electrokinetically and pressure driven flow in such soft nanochannels has never been addressed. The Poisson-Boltzmann equation is solved assuming the Debye-Huckel linearization for the low electric potential to provide us with analytical closed form solutions for the conservation equations. The conservation equations are solved to obtain the electric potential; velocity and temperature distributions in terms of governing dimensionless parameters. Also results for the Nusselt number are presented and discussed in detail.

When surface potentials (or the surface charges) are high, the exponential term on the right-hand side of the Poisson-Boltzmann equation cannot be linearized. This nonlinear regime is systematically studied for various porous media and various physicochemical conditions. As in the linear regime, the numerical data for the electroosmotic coefficient when made dimensionless are shown to follow the semianalytical solution derived for a plane Poiseuille flow. Therefore, this coefficient can be deduced either from the specific surface or from the permeability and the formation factor.

When clay soils are subjected to salt concentration gradients, various interrelated processes come into play. It is known that chemical osmosis induces a water flow and that a membrane potential difference develops that counteracts diffusive flow of solutes and osmotic flow of water. In this paper, we present the results of experiments on the influence of membrane potential on chemical osmotic flow and diffusion of solutes and we show how we are able to derive the membrane potential value from theory. Moreover, the simultaneous development of water pressure, salt concentration and membrane potential difference are simulated using a model for combined chemico-electroosmosis in clays. A new method for short-circuiting the clay sample is employed to assess the influence of electrical effects on flow of water and transport of solutes.

The dynamic electrophoretic response of a spherical dielectric particle suspended in an electrolyte solution to a step change in the applied electrics field is analytically studied. The electrical double layer surrounding the particle may have either a small but finite thickness or a very large thickness relative to the particle radius. For the case of electrophoresis of a particle with a thin double layer, the local electroosmotic velocity at the outer edge of the double layer evolving with time after the external field is imposed is used as an apparent slip boundary condition at the particle surface so that the unsteady equation of motion for the fluid flow outside the double layer is solved. Closed-form formulas for the transient electrophoretic mobility of the particle are derived as functions of relevant parameters. The results demonstrate that, when the double layer surrounding the particle is relatively thin, the normalized electrophoretic mobility at a given dimensionless time decreases monotonically with a decrease in the parameter kappaa, where kappa(-1) is the Debye screening length and a is the particle radius. When the double layer of the particle is relatively thick, the particle mobility can have magnitudes comparable to those for a particle with a thin double layer in the initial stage, but will become much smaller afterward. In general, the effect of the relaxation time for transient electrophoresis is negligible, regardless of the value of kappaa.

In this paper, we model frequency-dependent electro-osmosis in a capillary using the fully nonlinear Navier-Stokes equation (NSE) for viscous, incompressible, and homogeneous flow. We simulate the NSE using the finite element method, computing the solution for a closed capillary and compare it to the closed form solutions. It is confirmed that the second velocity zero crossing is dependent of the capillary radius. The distance of the zero velocity crossing decreases with decreasing capillary radius. It is also shown that the AC electro-osmosis causes a circulation of fluid within the capillary with low frequencies generating the greatest net flow.

This paper presents a mathematical model to describe a two-fluid electroosmotic pumping technique, in which an electrically non-conducting fluid is delivered by the interfacial viscous force of a conducting fluid; the latter is driven by electroosmosis. The electrical potential in the conducting fluid and the analytical solution of the steady two-fluid electroosmotic stratified flow in a rectangular microchannel was presented by assuming a planar interface between the two immiscible fluids. The effects of viscosity ratio, hold-up, concentration, and interfacial zeta potential are analyzed to show the potential feasibility of this technique.

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, ɛ μ , 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 ɛ μ values and has a concentration field similar to that of the smooth channels under high ɛ μ values.

This article presents an analysis of the frequency- and time-dependent electroosmotic flow in a closed-end rectangular microchannel. An exact solution to the modified Navier-Stokes equation governing the ac electroosmotic flow field is obtained by using the Green's function formulation in combination with a complex variable approach. An analytical expression for the induced backpressure gradient is derived. With the Debye-Hückel approximation, the electrical double-layer potential distribution in the channel is obtained by analytically solving the linearized two-dimensional Poisson-Boltzmann equation. Since the counterparts of the flow rate and the electrical current are shown to be linearly proportional to the applied electric field and the pressure gradient, Onsager's principle of reciprocity is demonstrated for transient and ac electroosmotic flows. The time evolution of the electroosmotic flow and the effect of a frequency-dependent ac electric field on the oscillating electroosmotic flow in a closed-end rectangular microchannel are examined. Specifically, the induced pressure gradient is analyzed under effects of the channel dimension and the frequency of electric field. In addition, based on the Stokes second problem, the solution of the slip velocity approximation is presented for comparison with the results obtained from the analytical scheme developed in this study.

An analytical model for electroosmotic flow rate, total pump current, and thermodynamic efficiency reported in a previous paper has been applied as a design guideline to fabricate porous-structure EO pumps. We have fabricated sintered-glass EO pumps that provide maximum flow rates and pressure capacities of 33 ml/min and 1.3 atm, respectively, at applied potential 100 V. These pumps are designed to be integrated with two-phase microchannel heat exchangers with load capacities of order 100 W and greater. Experiments were conducted with pumps of various geometries and using a relevant, practical range of working electrolyte ionic concentration. Characterization of the pumping performance are discussed in the terms of porosity, tortuosity, pore size, and the dependence of zeta potential on bulk ion density of the working solution. The effects of pressure and flow rate on pump current and thermodynamic efficiency are analyzed and compared to the model prediction. In particular, we explore the important tradeoff between increasing flow rate capacity and obtaining adequate thermodynamic efficiency. This research aims to demonstrate the performance of EOF pump systems and to investigate optimal and practical pump designs. We also present a gas recombination device that makes possible the implementation of this pumping technology into a closed-flow loop where electrolytic gases are converted into water and reclaimed by the system.

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.

The term electroosmotic flow refers to the bulk flow of an aqueous solution induced by the application of the electric field to the zeta potential. The characteristics of EOF in a microchannel depend upon the nature of the zeta potential, i.e., whether it is uniform or nonuniform. In this study, the full Navier-Stokes equation and the Nernst-Planck equation are used to model the change in EOF characteristics that occur when a step change in zeta potential is applied. It is found that the thickness of the electrical double layer gradually increases downstream from the location at which the zeta potential is increased. The results indicate that a step change in zeta potential causes a significant variation in the velocity profile and in the pressure distribution.

Displacement of one electrolyte solution by another in a microchannel is required in many biolab chip devices. The objective of this paper is to develop a better understanding of the displacement process between two electrolyte solutions under an applied electric field in a cylindrical microchannel in terms of the traveling distance of the interface between these two electrolyte solutions. In order to develop a general model to predict the location of the interface, two different situations are considered; one model assumes the presence of a sharp interface between the two solutions and the other model considers a mixing zone between the two solutions. Carefully conducted experiments were carried out to obtain the current–time relationship, which is used in the model to predict the location of the interface. In these experiments, deionized ultrafiltered water (DIUF water), 10 mM KCl, 0.1 mM KCl, and 0.1 mM LaCl3 solutions were used as the testing liquids. Polyamide-coated silica capillary tubes of internal diameter 100 μm and length 10 cm were employed in this study. The relationship between traveled distance of the interface and time was predicted by a developed model based on the measured current–time relationship for such a displacement process under a constant applied electric field. The characteristics of the nonlinear change of the traveling distance with the time were also discussed in this paper.

This paper presents a theory for frequency-dependent electroosmosis. It is shown that for a closed capillary the electroosmosis frequency-dependent ratio of DeltaV/DeltaP is constant with increasing frequency until inertial effects become prevalent, at which time DeltaV/DeltaP starts to decrease with increasing frequency. The frequency response of the electroosmosis coupling coefficient is shown to be dependent on the capillary radius. As the capillary radius is made smaller, inertial effects start to occur at higher frequencies. As part of this paper, frequency-dependent electroosmosis is compared to frequency-dependent streaming potentials. In this comparison it is shown that inertial effects start to become more prevalent at higher frequencies for the closed capillary frequency-dependent electroosmosis case than for the frequency-dependent streaming potential case in the same capillary. It is also shown that this difference is due to a second viscosity (transverse) wave that emanates from the velocity zero within the capillary for the electroosmosis case. The second viscosity wave superposes with the viscosity wave that emanates from wall of the capillary to effectively reduce the hydraulic radius of the capillary. Data are presented for a 0.127-mm capillary to support the findings in this paper.

The electroosmotic flow through an annulus is analyzed under the situation when the two cylindrical walls carry high zeta potentials. The analytical solutions for the electric potential profile and the electroosmotic flow field in the annulus are obtained by solving the Poisson-Boltzmann equation and the Stokes equation under an analytical scheme for the hyperbolic sine function. A mathematical expression for the average electroosmotic velocity is derived in a fashion similar to the Smoluchowski equation. Hence, a correction formula is introduced to modify the Smoluchowski equation, taking into account contributions due to the finite thickness of the electric double layer (EDL) and the geometry ratio-dependent correction. Specifically, under a circumstance when the two annular walls are oppositely charged, the flow direction can be determined from the sign of such correction formula, and there exists a zero-velocity plane inside the annulus. With the assumption of large electrokinetic diameters, the location of the zero-velocity plane can be estimated from the analytical expression for the velocity distribution. In addition, the characteristics of the electroosmotic flow through the annulus are discussed under the influences of the EDL parameters and geometric ratio of the inner radius to the outer radius of the annulus.

The steady diffusioosmotic and electroosmotic flows of an electrolyte solution in the fibrous porous medium constructed by a homogeneous array of parallel charged circular cylinders are analyzed under conditions of small Peclet and Reynolds numbers. The imposed electrolyte concentration gradient or electric field is constant and can be oriented arbitrarily with respect to the axes of the cylinders. The thickness of the electric double layers surrounding the cylinders is assumed to be small relative to the radius of the cylinders and to the gap width between two neighboring cylinders, but the polarization effect of the diffuse ions in the double layers is incorporated. Through the use of a unit cell model, the appropriate equations of conservation of the electrochemical potential energies of ionic species and the fluid momentum are solved for each cell, in which a cylinder is envisaged to be surrounded by a coaxial shell of the fluid. Analytical expressions for the diffusioosmotic and electroosmotic velocities of the bulk electrolyte solution as functions of the porosity of the ordered array of cylinders are obtained in closed form for various cases. Comparisons of the results of the cell model with different conditions at the outer boundary of the cell are made. In the limit of maximum porosity, these results can be interpreted as the diffusiophoretic and electrophoretic velocities of an isolated circular cylinder caused by the imposed electrolyte concentration gradient or electric field.

The hydrodynamic problem of electroosmotic flow in a cylindrical capillary with random zeta potential is solved in the limit of small Deybe length and low Reynolds number. Averages are defined over multiple experiments and the mean axial velocity is found to be a plug flow. The variance of the velocity exhibits parabolic-like variation across the capillary. Average concentrations of samples transported by the flow are approximated by defining an effective diffusivity coefficient. Theoretical formulas for the average concentration are supported by numerical experiments.

The transient aspects of electroosmotic flow in a slit microchannel are studied. Exact solutions for the electrical potential profile and the transient electroosmotic flow field are obtained by solving the complete Poisson-Boltzmann equation and the Navier-Stokes equation under an analytical approximation for the hyperbolic sine function. The characteristics of the transient electroosmotic flow are discussed under influences of the electric double layer and the geometric size of the microchannel.

The entry flow induced by an applied electrical potential through microchannels between two parallel plates is analyzed in this work. A nonlinear, two-dimensional Poisson equation governing the applied electrical potential and the zeta potential of the solid–liquid boundary and the Nernst–Planck equation governing the ionic concentration distribution are numerically solved using a finite-difference method. The applied electrical potential and zeta potential are unified in the Poisson equation without using linear superposition. A body force caused by the interaction between the charge density and the applied electrical potential field is included in the full Navier–Stokes equations. The effects of the entrance region on the fluid velocity distribution, charge density boundary layer, entrance length, and shear stress are discussed. The entrance length of the electroosmotic flow is longer than that of classical pressure-driven flow. The thickness of the electrical double layer (EDL) in the entry region is thinner than that in the fully developed region. The change of velocity profile is apparent in the entrance region, and the axial velocity profile is no longer flat across the channel height when the Reynolds number is large.

The characteristics of electroosmotic flow in a cylindrical microchannel with nonuniform zeta potential were investigated in this paper. The Poisson–Boltzmann equation and momentum equation were used to model the electrical double-layer field and the flow field. The numerical results show the distorted electroosmotic velocity profiles resulting from the axial variation of the zeta potential. Also, the influences of the unequal section size and the direction of the zeta potential change on the velocity profile, the induced pressure distribution, and the volumetric flow rate are discussed in this paper. The simulation results revealed possible effects of bioadhesion in microchannels on the electroosmotic flow in biochip devices.

Displacing one electrolyte solution with another solution in a microchannel is often required in many biomedical lab-on-a-chip devices. This paper discusses both theoretical and experimental studies of electroosmotic flow in a capillary with one electrolyte solution displacing another solution. A theoretical model was developed to predict the electroosmotic flow displacing process. This model considered the mixing process between the two different solutions and the induced pressure gradient in the capillary due to the different electrolyte solutions and hence the different electrokinetic conditions in different sections of the capillary. In the experiments, deionized ultrafiltered water, 10−2 M KCl solution, 10−4 M KCl solution, and 10−4 M LaCl3 solution were used as the testing fluid. Polyamide-coated silica capillary tubes 100 μm in internal diameter and 10 cm in length were used in this study. The nonlinear change of the current with time was found during such a displacing process under a constant applied electrical field. A good agreement between the experimentally measured current change and the model prediction of the current change was found. The characteristics of the mixing process are also discussed in the paper.

The transient response of electrolyte solutions in a narrow capillary tube or slit to a step change in the applied electric field and/or pressure gradient is analytically studied. The electric double layer adjacent to the charged capillary wall may have an arbitrary thickness relative to the capillary radius. The electrostatic potential distribution on a cross section of the capillary is developed by solving the Poisson–Boltzmann equation, and the fluid velocity profile evolving with time after the external field is imposed is obtained from the analytical solution of a modified Navier–Stokes equation. Closed-form formulas for the transient flow rate, electro-osmotic velocity, electric current, and streaming potential in the capillary are also derived as functions of relevant parameters. The results demonstrate that the behavior of the transient electrokinetic flow in a capillary tube and in a capillary slit is similar; however, the rate of evolution of the flow in a tube with time is faster by a factor of about 2 than that in a slit with its half thickness equal to the tube radius.

The electroosmotic flow induced by an applied electrostatic potential field through microchannels between two parallel plates and a 90 degrees bend is analyzed in this work. A nonlinear, two-dimensional Poisson-Boltzmann equation governing the electrical double-layer field and the Laplace equation governing the electrostatic field distribution in microchannels are numerically solved using a finite-difference method. A body force caused by the interaction between the electrical double-layer field and the applied electrostatic field is included in the full Navier-Stokes equations. The effects of the electrical double-layer field and the applied electrostatic field on the fluid velocity distribution, pressure drop, and skin friction are discussed. A small pressure drop along the parallel plates is detected, although it is always neglected in the literature. Pressure is not a constant across the channel height. The axial velocity profile is no longer flat across the channel height when the Reynolds number is large. A separation bubble is detected near the 90 degrees junction when the Reynolds number is large. Copyright 2001 Academic Press.

Mass-transfer systems based on electrokinetic phenomena (i.e., capillary electrochromatography (CEC)) have shown practical potential for becoming powerful separation methods for the biotechnology and pharmaceutical industries. A dynamic mathematical model, consisting of the momentum balance and the Poisson equations, as well as the unsteady-state continuity expressions for the cation and anion of the background electrolyte and of a positively charged analyte (adsorbate), is constructed and solved to determine quantitatively the electroosmotic velocity, the electrostatic potential, the concentration profiles of the charged species in the double layer and in the electroneutral core region of the fluid in the interstitial channels for bulk flow in the packed chromatographic column, and the axial current density profiles as the adsorbate adsorbs onto the negatively charged fixed sites on the surface of the nonporous particles packed in the chromatographic column. The frontal analysis mode of operation is simulated in this work. The results obtained from model simulations provide significant physical insight into and understanding of the development and propagation of the dynamic profile of the concentration of the adsorbate (analyte) and indicate that sharp, highly resolved adsorption fronts and large amounts of adsorbate in the adsorbed phase for a given column length can be obtained under the following conditions: (i) The ratio, gamma(2, 0), of the electroosmotic velocity of the mobile liquid phase at the column entrance after the adsorption front has passed the column entrance to the electrophoretic velocity of the anion is very close to -1. The structure of the equations of the model and model simulations indicate that a stable adsorption front cannot develop when gamma(2, 0) is less than -1 unless the value of the mobility of the cation is less than the value of the mobility of the analyte, which may be a rare occurrence in practical CEC systems. (ii) The ratio of the mobility of the cation to the mobility of the analyte is less than two orders of magnitude. This effect becomes more significant as the value of the equilibrium adsorption constant, K(A, 3), of the analyte increases. (iii) The concentration of the analyte relative to the concentration of the cation is increased (feed solutions with less dilute concentrations of the analyte are employed). Therefore, to obtain good performance for CEC systems operated in the frontal analysis mode (well-resolved adsorption fronts and high adsorbate amounts in the adsorbed phase), one can choose an electrolyte whose cation has a mobility that is not more than one or two orders of magnitude greater than the mobility of the analyte and whose anion has a mobility such that the value of gamma(2, 0) is close to -1; one can then bring the value of gamma(2, 0) closer to -1 by decreasing the particle diameter, d(p), and/or making the value of the surface charge density, delta(0), of the particles more negative (in effect, making the value of the zeta potential, zeta(p), at the surface of the particles more negative at time t=0) to change the value of the velocity,upsilon(x)|(x=0), of the electroosmotic flow (EOF) at the column entrance (upsilon(x)|(x=0) is determined after the adsorption front has passed the column entrance). This approach could provide conditions in the column that avoid overloading of the adsorbate. One can obtain faster breakthrough times at the sacrifice of resolution and utilization of the adsorptive capacity of the packed bed if one employs a cation whose mobility is very large relative to the mobility of the analyte and/or an anion that provides a value of gamma(2, 0) significantly greater than -1. If it is possible, one can increase the concentration of the analyte in the feed stream to avoid sacrificing resolution and adsorptive capacity of the packed bed and still decrease the time at which breakthrough occurs. Also, the dynamic behavior of the axial current density, i(x), profiles indicates that the magnitude of i(x) and/or the change in the value of i(x) across the adsorption front could serve as a measurement for the rate of propagation of the adsorption front through the column. Furthermore, the effect of the decreased magnitude of the velocity of the EOF in the region of the column where the analyte is present in the adsorbed phase could act to decrease the effect of tailing when CEC systems are operated in the pulse injection mode (analytical electrochromatography) because the higher velocity of the fluid upstream of the migrating adsorption zone may compress the tail of the peak. Copyright 2001 Academic Press.

The electro-osmosis through an annulus is investigated. The electric potential and flow velocity profile are obtained by solving the linearized Poisson-Boltzmann equation and the Stokes equation. Both the thin and thick double layer limits are analyzed. Under the condition of thin double layer, the electro-osmotic mobility can be described by the Helmholtz-Smoluchowski equation with a geometry-dependent correction factor. There exist net flows even for zero area-averaged surface charge density due to the curvature differences between the inner and outer walls. The flow direction is determined by the sign of the charge on the inner cylinder. We also found that under certain circumstances the flow direction in an annulus is opposite to that in a capillary with the same sign of the net charge. Copyright 2000 Academic Press.

In this paper we have outlined a simple procedure for determining zeta potential, zeta(0), and surface conductance, lambda(s), based on current monitoring in electro-osmosis. In these experiments, the average velocity was determined by measuring the amount of time required to completely displace a solution by another solution in the capillary tube. The average velocity during electro-osmosis was found to be independent of capillary size, although it was dependent on the electrolyte concentration and applied electrical field. The measured values of the zeta potential, zeta(0), were found to be independent of capillary size and the applied field, while zeta(0) is strongly dependent on the electrolyte concentration. Calculations of the specific surface conductivity lambda(s) based on current measurements reveal a relationship between lambda(s) and capillary size, in agreement with the results reported in the literature. Copyright 2000 Academic Press.

The electroosmotic flow of a general electrolyte solution through a fibrous medium is modeled theoretically taking the effect of double-layer polarization into account. The result obtained is applicable to an arbitrary level of electrical potential. We show that if the effect of double-layer polarization is neglected using the linearized Poisson-Boltzmann equation will underestimate electroosmotic velocity. The deviation becomes inappreciable, however, if kappaa is either very large or very small, kappa and a being, respectively, the reciprocal Debye length and the radius of a fiber. If the surface potential is high, the variation of electroosmotic velocity as a function of kappaa may exhibit a local maximum and a local minimum, and the larger the porosity of the fibrous medium the lower the level of surface potential for the local extremals to occur. If kappaa is small, the effect of surface potential on the electroosmotic velocity is more significant than that of double-layer polarization, and the reverse is true if kappaa is large. Copyright 2000 Academic Press.

The effect of the magnetic field on the electrokinetic transport coefficients (permeability coefficient and electro-osmotic permeability coefficient) of water and aqueous solutions of mercuric chloride and glycine through a sintered disc impregnated with cellulose acetate at different potentials, concentrations, and magnetic fields varying up to 21 kg/cm(2) are reported at 308.15 K. The phenomenological coefficients characterizing the electro-osmotic flow and the membrane characteristics are also estimated for the various solutions with the object of determining the efficiencies of electrokinetic energy conversion and zeta potential. The effect of magnetic field has been attributed to the molecular orientation of dipoles in solutions and to the change in the structure of the membrane. Copyright 1999 Academic Press.

Nonlinear dynamic behavior and electrokinetic oscillations have been investigated for the membrane systems (a) 0.1 N NaCl/KCl parallel Millipore filter parallel 0.01 N NaCl/KCl; (b) 0.1 N NaCl/KCl parallel Whatman Inorganic filter parallel 0.01 N NaCl/KCl; and (c) 0.1 N NaCl/KCl parallel silver-coated filter parallel 0.01 N NaCl/KCl, from the viewpoint of testing the theories for the phenomena and elucidating the mechanism. To achieve these objectives, studies on hydrodynamic permeability, electroosmotic permeability, bistability, and electrokinetic oscillations were undertaken. Relaxation time for buildup and decay of electroosmotic pressures was experimentally determined. Bistability was not observed showing that it is not a prerequisite for oscillations and nonlinear relations between (J(v))(Deltarho=0) and Deltaphi; involving cubic or higher-order terms are necessary for bistability. The oscillations were studied at different current strengths. The period is found to be independent of current, while amplitude A is found to be linearly related to current I which is the bifurcation parameter. The bifurcation point occurs at approximately 0.4 mA. Studies have also been made with membranes of different pore size that show that amplitude increases with increase in pore size of the membranes. The validity of the two-variable model of Teorell was examined by comparing the experimental results with computer simulation based on parameters determined experimentally. Theory does not meet expectation and the results suggest that modification of theory is needed. The weakness of the theory has been critically examined. Copyright 1999 Academic Press.

The electroosmotic flux through a cation-exchange membrane has been obtained in different situations. From these measurements, the apparent electroosmotic permeability, W, of a cation-exchange membrane has been determined as a function of the temperature, T, and the stirring rate, v, of the solutions. In all the experimental situations studied, W decreases when v increases, while it can increase or decrease with T depending on the temperature range considered. For this last reason, the (T,W) curves show a minimum whose value and position depend on the experimental conditions established. The influence of the concentration polarization effect in the value of W and in its dependence with v and T is studied and quantified. Copyright 1999 Academic Press.

This note provides a few comments on the conditions required for similitude between velocity and electric field in electroosmotic flows. The velocity fields of certain electroosmotic flows with relatively thin electric double layers (EDLs) are known to be irrotational in regions outside of the EDL. Under restricted conditions, the velocity field, V ¯ , can be expressed in terms of the electric field, E ¯ , as V ¯ = c E ¯ , where c is a scalar constant. The irrotationality solution is certainly unique and exact for Stokes flow, but may not be stable (or unique) for flows with Reynolds numbers significantly greater than unity.

We derive an expression for the L 12 coefficient responsible for electroosmosis. For constant surface charge, we write this as an extension of the Helmholtz–Smoluchowski result. The leading order correction introduces a geometric length scale similar to the Λ parameter.

The current research of induced-charge electroosmotic flow (ICEOF) is mostly confined to systems with ideally or fully polarizable surfaces (e.g., metal). However, most materials in nature have various degrees of polarizability, which directly affects the induced charges and subsequently the induced-charge electroosmotic flow. This paper studied the effect of the polarizability of the materials on the ICEOF. An analytical expression of the induced potential on the surface of a dielectric particle in a uniform electrical field was derived. Three-dimensional transient numerical simulations of the ICEOF and the motion of dielectric particles were performed to study the effect of the polarizability. Simulation results show that the transportation of the dielectric particle in a microchannel is not affected by the polarizability of the particle; however, the interaction of two dielectric particles is sensitive to the polarizability of the particles.