Electrokinetic flow in an elliptic microchannel covered by ion-penetrable membrane

The electrokinetic flow of an electrolyte solution in an elliptical microchannel covered by an ion-penetrable, charged membrane layer is examined theoretically. The present analysis extends previous results in that a two-dimensional problem is considered, and the system under consideration simulates the flow of a fluid, for example, in a microchannel of biological nature such as vein. The electroosmostic volumetric flow rate, the total electric current, the streaming potential, and the electroviscous effect of the system under consideration are evaluated. We show that, for a constant hydraulic diameter, the variations of these quantities as a function of the aspect ratio of a microchannel may have a local minimum or a local maximum at a medium level of ionic strength, which depends on the thickness of the membrane layer. For a constant cross-sectional area, the electroosmostic volumetric flow rate, the total electric current, and the streaming potential increase monotonically with the increase in the aspect ratio, but the reverse is true for the electroviscous effect.     

Electrokinetic flow through an elliptical microchannel: effects of aspect ratio and electrical boundary conditions

The electrokinetic flow of an electrolyte solution through an elliptical microchannel is studied theoretically. The system under consideration simulates the flow of a fluid, for example, in vein. We show that, for a constant cross-sectional area, both the electroosmotic volumetric flow rate and the streaming potential increase monotonically with an increase in the aspect ratio, and both the total electric current and the electroviscous effect may exhibit a local minimum as the aspect ratio varies. Also, for a constant average potential on the channel wall, the difference between the results based on three kinds of boundary conditions, which include constant surface charge, constant surface potential, and charge-regulated surface, is inappreciable if the hydraulic diameter is larger than 1 mum.     

Electrokinetic flow in a planar slit covered by an ion-penetrable charged membrane

The electrokinetic flow of an electrolyte solution in a planar slit covered by an ion-penetrable charged membrane layer is analyzed theoretically. An approximate analytical expression for the spatial variation in the electrical potential is derived, and the electroosmotic velocity, the total electric current, and the streaming potential of the system under consideration are evaluated. The effects of epsilon' (relative permittivity of liquid phase/relative permittivity of membrane layer), eta' (viscosity of liquid phase/viscosity of membrane layer) and the valence of anions (coions) on the volumetric flow rate and total current are examined. We show that the effect of the valence of cations (counterions) on the volumetric flow rate is less significant than that of epsilon' and that of eta'. However, the effect of epsilon' on the total current is less significant than that of the valence of cations and that of eta'. The variation of total current as a function of ionic strength is found to have a local minimum, regardless of whether a pressure gradient is applied or not. The absolute streaming potential has a local maximum as the concentration of fixed charge varies, which was not found in previous studies.     

Oscillating laminar electrokinetic flow in infinitely extended rectangular microchannels

This paper has addressed analytically the problem of laminar flow in microchannels with rectangular cross-section subjected to a time-dependent sinusoidal pressure gradient and a sinusoidal electric field. The analytical solution has been determined based on the Debye-Hückel approximation of a low surface potential at the channel wall. We have demonstrated that Onsager's principle of reciprocity is valid for this problem. Parametric studies of streaming potential have shown the dependence of the electroviscous effect not only on the Debye length, but also on the oscillation frequency and the microchannel width. Parametric studies of electroosmosis demonstrate that the flow rate decreases due to an increase in frequency. The obtained solutions for both the streaming potential and electroosmotic flows become those for flow between two parallel plates in the limit of a large aspect ratio.     

Oscillating laminar electrokinetic flow in infinitely extended circular microchannels

This article addresses the problem of oscillating laminar electrokinetic liquid flow in an infinitely extended circular microchannel. Based on the Debye-Huckel approximation for low surface potential at the channel wall, a complex variable approach is used to obtain an analytical solution for the flow. The complex counterparts of the flow rate and the current are linearly dependent on the pressure gradient and the external electric field. This property is used to show that Onsager's principle of reciprocity continues to be valid (involving the complex quantities) for the stated problem. During oscillating pressure-driven flow, the electroviscous effect for a given value of the normalized reciprocal electrical double-layer (EDL) thickness is observed to attain a maximum at a certain normalized frequency. In general, an increasing normalized frequency results in a reduction of EDL effects, leading to (i). a volumetric flow rate in the case of streaming potential approaching that predicted by the theory without EDL effects, and (ii). a reduction in the volumetric flow rate in the case of electroosmosis.     

Theoretical analysis of a novel electrical field assisted membrane module comprising an array of microchannel units

A novel electrical field assisted membrane module consisting of an array of microchannel units, each microchannel unit comprised of a cylindrical pore and a charged ion-selective membrane layer, is analyzed theoretically. The governing equations for the flow and the electrical fields are solved analytically under the Debye-Huckel condition and the influences of the key parameters on the flow behavior of the system under consideration are investigated through numerical simulation. We show that for a fixed microchannel radius, the volumetric flow rate through a microchannel unit has a maximal value as the radius of the cylindrical pore varies. This maximum is independent of both the strength of the applied field and the density of the fixed charges in the membrane layer, but varies with the permittivity of the membrane layer.     

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.     

Transient streaming potential in a finite length microchannel

Pressure-driven flow of an electrolyte solution in a microchannel with charged solid surfaces induces a streaming potential across the microchannel. Such a flow also causes rejection of ions by the microchannel, leading to different concentrations in the feed and permeate reservoirs connecting the capillary, which forms the basis of membrane based separation of electrolytes. Modeling approaches traditionally employed to assess the streaming potential development and ion rejection by capillaries often present a confusing picture of the governing electrochemical transport processes. In this paper, a transient numerical simulation of electrochemical transport process leading to the development of a streaming potential across a finite length circular cylindrical microchannel connecting two infinite reservoirs is presented. The solution based on finite element analysis shows the transient development of ionic concentrations, electric fields, and the streaming potential over the length of the microchannel. The transient analysis presented here resolves several contradictions between the two types of modeling approaches employed in assessing streaming potential development and ion rejection. The simulation results show that the streaming potential across the channel is predominantly set up at the timescale of the developing convective transport, while the equilibrium ion concentrations are developed over a considerably longer duration.     

Residence time distribution for electrokinetic flow through a microchannel comprising a bundle of cylinders

The electrokinetic flow of an electrolyte solution through a microchannel that comprises a bundle of cylinders is investigated for the case of constant surface potential. The system under consideration is simulated by a unit cell model, and analytical expressions for the flow field and the corresponding residence time distribution under various conditions are derived. These results are readily applicable to the assessment of the performance of a microreactor such as that which comprises a bundle of optical fibers. Numerical simulations are conducted to investigate the influences of the key parameters, including the thickness of the double layer, the strength of the applied electric field, the magnitude of the applied pressure gradient, and the characteristic sizes of a microchannel, on the residence time distribution. We show that the following could result in a shorter residence time: thin double layer, strong applied electric field, large applied pressure gradient, and small number of cylinders. Based on the thickness of the double layer, criteria are proposed for whether the flow field can be treated as a laminar flow or as a plug flow, two basic limiting cases in reactor design.     

Exploring new scaling regimes for streaming potential and electroviscous effects in a nanocapillary with overlapping Electric Double Layers

In this paper, we unravel new scaling regimes for streaming potential and electroviscous effects in a nanocapillary with thick overlapping Electric Double Layers (EDLs). We observe that the streaming potential, for a given value of the capillary zeta (ζ) potential, varies with the EDL thickness and a dimensionless parameter R, quantifying the conduction current. Depending on the value of R, variation of the streaming potential with the EDL thickness demonstrates distinct scaling regimes: one can witness a Quadratic Regime where the streaming potential varies as the square of the EDL thickness, a Weak Regime where the streaming potential shows a weaker variation with the EDL thickness, and a Saturation Regime where the streaming potential ceases to vary with the EDL thickness. Effective viscosity, characterizing the electroviscous effect, obeys the variation of the streaming potential for smaller EDL thickness values; however, for larger EDL thickness the electroosmotic flow profile dictates the electroviscous effect, with insignificant contribution of the streaming potential.     

Determining the zeta-potential of ceramic microfiltration membranes using the electroviscous effect

The possibility of measuring the zeta-potentials of porous membranes using the electroviscous effect was investigated. The zeta-potential of Membralox^® ceramic microfiltration membranes was determined both with the newly developed electroviscous technique and by streaming potential measurements. It was found that the electroviscous technique provided a simple means of obtaining accurate values of zeta-potential, especially for higher zeta-potentials. The streaming potential measurements were found to be more suitable for the determination of the iso-electric point, i.e. the pH at which the zeta-potential is zero.The iso-electric points of new α-alumina, zirconia, and titania membranes were found to be 8.5, 8.0, and 6.3, respectively. Upon using the membranes and cleaning them with a detergent, the iso-electric point of the α-alumina membrane decreased to 6.5, and that of the zirconia membrane decreased to 5.2, while the iso-electric point of the titania membrane stayed virtually constant. Cleaning these membranes with a strong acid or base could not reverse the observed decreases in iso-electric point.     

Electroviscous effect on the streaming current in a pH-regulated nanochannel

Accurate and rapid estimation of the streaming current in nanochannels is crucial for the development of the nanofluidics based power generation apparatus. In this study, an analytical model is developed for the first time to examine the electroviscous effect on the streaming current/conductance in a pH-regulated nanochannel by considering practical effects of multiple ionic species, surface chemistry reactions, and the Stern layer. Predictions from the model are in good agreement with the experimental results of the streaming conductance in silica nanochannels available in the literature. The electroviscous effect could have a significant reduction of ca. 30% in the streaming conductance at medium pH and low salt concentration.     

Analysis of rotation-driven electrokinetic flow in microscale gap regions of rotating disk systems

In the present study, a novel theoretical model is developed for the analysis of rotating thermal-fluid flow characteristics in the presence of electrokinetic effects in the microscale gap region between two parallel disks under specified electrostatic, rotational, and thermal boundary conditions. The major flow configuration considered is a rotor-stator disk system. Axisymmetric Navier-Stokes equations with consideration of electric body force stemming from streaming potential are employed in the momentum balance. Variations of the fluid viscosity and permittivity with the local fluid temperature are considered. Between two disks, the axial distribution of the electric potential is determined by the Poisson equation with the concentration distributions of positive and negative ions obtained from Nernst-Planck equations for convection-diffusion of the ions in the flow field. Effects of disk rotation and electrostatic and thermal conditions on the electrokinetic flow and thermal characteristics are investigated. The electrohydrodynamic mechanisms are addressed with an interpretation of the coupling nature of the electric and flow fields. Finally, solutions with electric potential determined by employing nonlinear or linearized Poisson-Boltzmann equation and/or invoking assumptions of constant properties are compared with the predictions of the present model for justification of various levels of approximation in solution of the electrothermal flow behaviors in rotating microfluidic systems.     

Hydrodynamic control of the interface between two liquids flowing through a horizontal or vertical microchannel

We derive a Fourier expansion for estimating laminar hydrodynamic spreading when two immiscible liquids flow steadily, side by side, at low Reynolds number, in a horizontal microchannel of rectangular cross-section. When the aspect ratio of the height of the channel to its width is small this expansion can be truncated after a couple of terms and solved to yield the steady-state position of the boundary in terms of the constant volumetric flow rates, the aspect ratio and the viscosities of the two liquids. Our formula shows that we can control the location of the liquid boundary at the outlet of the microchannel by adjusting the relative volumetric flow rates of the two inlet streams. If the microchannel is oriented vertically and the volumetric flow rates are very low, the position of the boundary is also influenced by gravitational forces that depend on the relative densities of the two liquids. We have tested solutions to our analytical expansions by running computational fluid dynamics simulations and have found excellent agreement. When the aspect ratio is 0.01 the leading term in the Fourier expansion for the spreading has an accuracy of about 1%. This accuracy deteriorates to about 12% when the aspect ratio is 0.1. An understanding of these spreading mechanisms is fundamental to the design of T-sensors and related microfluidic devices.     

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.     

Quasi-reversible faradaic depolarization processes in the electrokinetics of the metal/solution interface

Bipolar faradaic depolarization of the metal/solution interface is quantitatively analyzed for the case where the solution is subject to lateral flow and contains a quasi-reversible redox couple. Transversal convective diffusion of the electroactive species and a position-dependent degree of reversibility of the interfacial electron-transfer (e.t.) reaction are among the primary features that govern depolarization. The spatial distributions of species concentrations and electric potential are numerically simulated. The system is characterized by nonlinear coupling between the transport (diffusion and flow) and the electric potential distribution under conditions of finite local currents. The resulting picture is that the reversibility of the e.t. reaction varies with position on the surface, with the highest reversibility downstream. This, in itself, generally leads to strongly asymmetric profiles of the faradaic current density along the surface. The impact on the electrokinetic properties of the interface is huge. For example, the steady-state streaming potential is depressed by the contribution from the bipolar faradaic process to the back current to an extent that varies from insignificant to complete, depending on the e.t. rate constant and concentrations of the electroactive species.     

Field-effect flow control in a polydimethylsiloxane-based microfluidic system.

The application of the field-effect for direct control of electroosmosis in a polydimethylsiloxane (PDMS)-based microfluidic system, constructed on a silicon wafer with a 2.0 microm electrically insulating layer of silicon dioxide, is demonstrated. This microfluidic system consists of a 2.0 cm open microchannel fabricated on a PDMS slab, which can reversibly adhere to the silicon wafer to form a hybrid microfluidic device. Aside from mechanically serving as a robust bottom substrate to seal the channel and support the microfluidic system, the silicon wafer is exploited to achieve field-effect flow control by grounding the semiconductive silicon medium. When an electric field is applied through the channel, a radial electric potential gradient is created across the silicon dioxide layer that allows for direct control of the zeta potential and the resulting electroosmotic flow (EOF). By configuring this microfluidic system with two power supplies at both ends of the microchannel, the applied electric potentials can be varied for manipulating the polarity and the magnitude of the radial electric potential gradient across the silicon dioxide layer. At the same time, the longitudinal potential gradient through the microchannel, which is used to induce EOF, is held constant. The results of EOF control in this hybrid microfluidic system are presented for phosphate buffer at pH 3 and pH 5. It is also demonstrated that EOF control can be performed at higher solution pH of 6 and 7.4 by modifying the silicon wafer surface with cetyltrimethylammonium bromide (CTAB) prior to assembly of the hybrid microfluidic system. Results of EOF control from this study are compared with those reported in the literature involving the use of other microfluidic devices under comparable solution conditions.     

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.     

Liquid transport in rectangular microchannels by electroosmotic pumping

The characteristics of electroosmotic flow in rectangular microchannels were investigated in this paper. A 2D Poisson–Boltzmann equation and the 2D momentum equation were used to model the electric double layer field and the flow field. The numerical solutions show significant influences of the channel cross-section geometry (i.e. the aspect ratio) on the velocity field and the volumetric flow rate. Also, the numerical simulation of the electroosmotic flow reveals how the velocity field and the volumetric flow rate depend on the ionic concentration, zeta potential, channel size and the applied electrical field strength.     

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

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