Generalized model for time periodic electroosmotic flows with overlapping electrical double layers

In this article, an analytical model is devised for analyzing time periodic electroosmotic flows through nanochannels within the continuum regime, without presuming the validity of the Boltzmann distribution of ionic charges. The charge density distributions are obtained from the conservation considerations of the individual ionic species and other thermochemical constraints and are subsequently utilized to derive the potential distribution within the electrical double layer (EDL). This, coupled with the Navier-Stokes equation, yields a closed-form expression of the time-dependent velocity field that is valid under overlapped EDL conditions. This expression is first validated in asymptotic limits of thin EDLs, for which closed form expressions have been benchmarked in the literature. Further analyses are carried out to bring out the influences of the frequency of the electrical field on the electroosmotic flow features in the presence of overlapped EDLs.     

Analysis of electroosmotic flow of power-law fluids in a slit microchannel

Electroosmotic flow of power-law fluids in a slit channel is analyzed. The governing equations including the linearized Poisson-Boltzmann equation, the Cauchy momentum equation, and the continuity equation are solved to seek analytical expressions for the shear stress, dynamic viscosity, and velocity distribution. Specifically, exact solutions of the velocity distributions are explicitly found for several special values of the flow behavior index. Furthermore, with the implementation of an approximate scheme for the hyperbolic cosine function, approximate solutions of the velocity distributions are obtained. In addition, a generalized Smoluchowski velocity is introduced by taking into account contributions due to the finite thickness of the electric double layer and the flow behavior index of power-law fluids. Calculations are performed to examine the effects of kappaH, flow behavior index, double layer thickness, and applied electric field on the shear stress, dynamic viscosity, velocity distribution, and average velocity/flow rate of the electroosmotic flow of power-law fluids.     

Electroosmotic Flow through an Annulus

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.     

Frequency-dependent laminar electroosmotic flow in a closed-end rectangular microchannel

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.     

Flow behavior of periodical electroosmosis in microchannel for biochips

This paper presents an analytical solution for periodical electroosmotic flows in two-dimensional uniform microchannel based on Poisson-Boltzmann equations for electric double layer (EDL) and Navier-Stokes equation for incompressible viscous fluid. Analytical results indicate that the velocity of periodical electroosmosis strongly depends on Reynolds number Re=omegah(2)/nu, as well as on EDL properties and applied electric field. Slip velocity of EDL decreases as the Reynolds number increases. Electroosmotic velocity outside the EDL decreases, and lag phase angle of velocity increases as distance away from the channel wall increases. A wavelike velocity profile across the channel is found. An asymptotic solution for low Reynolds number is given in this paper. Periodical electroosmosis with low Reynolds has same velocity amplitude and a pluglike velocity profile as that of steady electroosmosis. Based on Debye-Hückel approximation, this paper also obtains a solution of periodical electroosmosis applicable to cases where the thickness of EDL is of the same order as half of channel width.     

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

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

Electroosmotic flow in a water column surrounded by an immiscible liquid

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 ?-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 ?-potential at the interface of the two fluids. The higher values of ?-potential also increase the slip velocity at the interface of the two fluids. For the same applied electric field and the same ?-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.     

Transient analysis of electroosmotic flow in a slit microchannel

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.     

Comments on the conditions for similitude in electroosmotic flows

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 =cE , 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.     

Joule heating induced transient temperature field and its effects on electroosmosis in a microcapillary packed with microspheres

The Joule heating induced transient temperature field and its effect on the electroosmotic flow in a capillary packed with microspheres is analyzed numerically using the control-volume-based finite difference method. The model incorporates the coupled momentum equation for the electroosmotic velocity, the energy equations for the Joule heating induced temperature distributions in both the packed column and the capillary wall, and the mass and electric current continuity equations. The temperature-dependent physical properties of the electrolyte solution are taken into consideration. The characteristics of the Joule heating induced transient development of temperature and electroosmotic flow fields are studied. Specifically, the simulation shows that the presence of Joule heating causes a noticeable axial temperature gradient in the thermal entrance region and elevates a significant temperature increment inside the microcapillary. The temperature changes in turn greatly affect the electroosmotic velocity by means of the temperature-dependent fluid viscosity, dielectric constant, and local electric field strength. Furthermore, the model predicts an induced pressure gradient to counterbalance the axial variation of the electroosmotic velocity so as to maintain the fluid mass continuity. In addition, under specific conditions, the present model is validated by comparing with the existing analytical model and experimental data from the literature.     

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

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

Transient electroosmotic flow induced by AC electric field in micro-channel with patchwise surface heterogeneities

This paper investigates two-dimensional, time-dependent electroosmotic flow driven by an AC electric field via patchwise surface heterogeneities distributed along the micro-channel walls. The time-dependent flow fields through the micro-channel are simulated for various patchwise heterogeneous surface patterns using the backwards-Euler time stepping numerical method. Different heterogeneous surface patterns are found to create significantly different electrokinetic transport phenomena. The transient behavior characteristics of the generated electroosmotic flow are then discussed in terms of the influence of the patchwise surface heterogeneities, the direction of the applied AC electric field, and the velocity of the bulk flow. It is shown that the presence of oppositely charged surface heterogeneities on the micro-channel walls results in the formation of localized flow circulations within the bulk flow. These circulation regions grow and decay periodically in phase with the applied periodic AC electric field intensity. The location and rotational direction of the induced circulations are determined by the directions of the bulk flow velocity and the applied electric field.     

Effects of overlapping electric double layer on mass transport of a macro‐solute across porous wall of a micro/nanochannel for power law fluid

Effects of overlapping electric double layer and high wall potential on transport of a macrosolute for flow of a power law fluid through a microchannel with porous walls are studied in this work. The electric potential distribution is obtained by coupling the Poisson's equation without considering the Debye–Huckel approximation. The numerical solution shows that the center line potential can be 16% of wall potential at pH 8.5, at wall potential −73 mV and scaled Debye length 0.5. Transport phenomena involving mass transport of a neutral macrosolute is formulated by species advective equation. An analytical solution of Sherwood number is obtained for power law fluid. Effects of fluid rheology are studied in detail. Average Sherwood number is more for a pseudoplastic fluid compared to dilatant upto the ratio of Poiseuille to electroosmotic velocity of 5. Beyond that, the Sherwood number is independent of fluid rheology. Effects of fluid rheology and solute size on permeation flux and concentration of neutral solute are also quantified. More solute permeation occurs as the fluid changes from pseudoplastic to dilatant.     

Electroosmotic flow in microchannels with arbitrary geometry and arbitrary distribution of wall charge

General solutions are developed for direct current (DC) and alternating current (AC) electroosmotic flows in microfluidic channels with arbitrary cross-sectional geometry and arbitrary distribution of wall charge (zeta potential). The applied AC electric field can also be of arbitrary waveform. By proposing a nondimensional time scale varpi defined as the ratio of the diffusion time of momentum across the electric double-layer thickness to the period of the applied electric field, we demonstrate analytically that the Helmholtz-Smoluchowski electroosmotic velocity is an appropriate slip condition for AC electroosmotic flows in typical microfluidic applications. With this slip condition approach, electroosmotic flows in rectangular and asymmetric trapezoidal microchannels with nonuniform wall charge, as examples, are investigated. The unknown constants in the proposed general solutions are numerically determined with a least-squares method through matching the boundary conditions. We find that the wall charge affects significantly the electroosmotic flow while the channel geometry does not. Moreover, the flow feature is characterized by another nondimensional time scale Omega defined as the ratio of the diffusion time of momentum across the channel hydraulic radius to the period of the applied electric field. The onset of phase shift between AC electroosmotic velocity and applied electric field is also examined analytically.     

Electrochemistry of capillary systems with narrow pores. II. Electroosmosis

Manegold and Solf have reported systematic deviations of the electroosmotic properties of collodion membranes with narrow pores from predictions based on the Helmholtz–Smoluchowski model. To interpret the electroosmotic data quantitatively it is necessary to replace the assumption of the Helmholtz–Smoluchowski model that the thickness of the electric double layer is small compared with the pore radius by a new assumption. We have assumed that the counter ions are distributed homogeneously in the pore fluid. In Part I of this series of contributions, equations have been given describing the electroosmotic properties of a membrane with narrow pores based on the new assumption. These equations are derived here in detail and are applied to an analysis of the experimental data given by Manegold and Solf.     

Electrophoresis of solid particles at large Peclet numbers

A theory of concentration polarization of a thin electrical double layer (DL) on a spherical particle is developed for the regime of large Peclet numbers which is realized in strong electric fields. In this regime, the concentration field arising outside DL is estimated under influence of diffusion and convection. According to the theory developed, polarization of DL at large Peclet numbers causes a change in the Stern potential, the formation of a dipole moment and the long-range potential. The diffuse layer deviates strongly from spherical symmetry and electroneutrality, and the screen of the surface charge is provided not only by the diffuse atmosphere but also by the charge induced in the convective-diffusion layer. The effect of electric field on the induced charge gives rise to the additional electroosmotic slip, that was called "secondary electroosmosis". Thus, a nonlinear additional term for the Smoluchowski formula of electrophoretic velocity is based on the changes of zeta-potential and on the secondary electroosmotic slip. The comparison of theory with experimental results revealed considerable fitting.     

Electrostatic potential and electroosmotic flow in a cylindrical capillary filled with symmetric electrolyte: analytic solutions in thin double layer approximation

The electrostatic potential in a capillary filled with electrolyte is derived by solving the nonlinear Poisson-Boltzmann equation using the method of matched asymptotic expansions. This approach allows obtaining an analytical result for arbitrary high wall potential if the double layer thickness is smaller than the capillary radius. The derived expression for the electrostatic potential is compared to numerical solutions of the Poisson-Boltzmann equation and it is shown that the agreement is excellent for capillaries with radii greater or equal to four times the electrical double layer thickness. The knowledge of the electrostatic potential distribution inside the capillary enables the derivation of the electroosmotic velocity flow profile in an analytical form. The obtained results are applicable to capillaries with radii ranging from nanometers to micrometers depending on the ionic strength of the solution.     

Diffusioosmotic flow in rectangular microchannels

The diffusioosmosis of an electrolyte solution inside a uniformly charged rectangular channel at steady locally developed conditions is the subject of this study. Utilizing a finite element based numerical procedure, we try to estimate the errors incurred by modeling the actual rectangular geometry of typical microchannels as a slit. We demonstrate that the flow pattern and direction are generally dependent upon the width‐to‐height ratio of the channel. Such a finding, besides showing the ineffectiveness of the slit geometry in representing a rectangular channel of small aspect ratio, informs us of another mechanism of controlling the diffusioosmotic flow. Inspections of the mean velocity reveal that, although it drastically grows by increasing the aspect ratio at smaller values of this parameter, no significant change is observed when the aspect ratio is 5 or higher. The same trend is observed when EDL is shrunk and is considered as a basis for the introduction of a slip‐like velocity, similar to the concept of the Helmholtz–Smoluchowski electroosmotic velocity, which will be of high practical importance when dealing with a micronsized channel. Because of its significance, an expression is presented for this slip velocity utilizing the curve fitting of the results, assuming a typical Peclet number.     

Electrochemistry of capillary systems with narrow pores. II. Electroosmosis

Manegold and Solf have reported systematic deviations of the electroosmotic properties of collodion membranes with narrow pores from predictions based on the Helmholtz–Smoluchowski model. To interpret the electroosmotic data quantitatively it is necessary to replace the assumption of the Helmholtz–Smoluchowski model that the thickness of the electric double layer is small compared with the pore radius by a new assumption. We have assumed that the counter ions are distributed homogeneously in the pore fluid. In Part I of this series of contributions, equations have been given describing the electroosmotic properties of a membrane with narrow pores based on the new assumption. These equations are derived here in detail and are applied to an analysis of the experimental data given by Manegold and Solf.     

Start‐up of electrophoresis of an arbitrarily oriented dielectric cylinder

An analytical study is presented for the transient electrophoretic response of a circular cylindrical particle to the step application of an electric field. The electric double layer adjacent to the particle surface is thin but finite compared with the radius of the particle. The time‐evolving electroosmotic velocity at the outer boundary of the double layer is utilized as a slip condition so that the transient momentum conservation equation for the bulk fluid flow is solved. Explicit formulas for the unsteady electrophoretic velocity of the particle are obtained for both axially and transversely applied electric fields, and can be linearly superimposed for an arbitrarily‐oriented applied field. If the cylindrical particle is neutrally buoyant in the suspending fluid, the transient electrophoretic velocity is independent of the orientation of the particle relative to the applied electric field and will be in the direction of the applied field. If the particle is different in density from the fluid, then the direction of electrophoresis will not coincide with that of the applied field until the steady state is attained. The growth of the electrophoretic mobility with the elapsed time for a cylindrical particle is substantially slower than for a spherical particle.