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.     

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

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

Estimation of the zeta potential and the dielectric constant using velocity measurements in the electroosmotic flows

In this paper we develop a method for the determination of the zeta potential zeta and the dielectric constant epsilon by exploiting velocity measurements of the electroosmotic flow in microchannels. The inverse problem is solved through the minimization of a performance function utilizing the conjugate gradient method. The present method is found to estimate zeta and epsilon with reasonable accuracy even with noisy velocity measurements.     

Estimation of inhomogeneous zeta potential using velocity measurements of EOF

We have developed a method of estimating the zeta potential distribution along the microchannel wall using velocity measurements of the EOF. The relevant inverse problem is solved through the minimization of a performance function utilizing the conjugate gradient method. Employing a set of simulated velocity measurements, which is constructed by adding random noise to the computed exact velocity, the present method is found to estimate the distribution of the zeta potential along the channel wall with reasonable accuracy.     

Analytical treatment of electrokinetic microfluidics in hydrophobic microchannels

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

Effects of surface heterogeneity on flow circulation in electroosmotic flow in microchannels

The characteristics of electroosmotic flow in a cylindrical microchannel with non-uniform zeta potential distribution are investigated in this paper. Two-dimensional full Navier–Stokes equation is used to model the flow field and the pressure field. The numerical results show the distorted electroosmotic velocity profiles and various kinds of flow circulation resulting from the axial variation of the zeta potential. The influences of heterogeneous patterns of zeta potential on the velocity profile, the induced pressure distribution and the volumetric flow rate are discussed in this paper. This work shows that using either heterogeneous patterns of zeta potential or a combination of a heterogeneous zeta potential distribution and an applied pressure difference over the channel can generate local flow circulations and hence provide effective means to improve the mixing between different solutions in microchannels.     

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.     

Numerical study of electroosmotic slip flow of fractional Oldroyd-B fluids at high zeta potentials

In this paper, an investigation of the electroosmotic flow of fractional Oldroyd-B fluids in a narrow circular tube with high zeta potential is presented. The Navier linear slip law at the walls is considered. The potential field is applied along the walls described by the nonlinear Poisson–Boltzmann equation. It's worth noting here that the linear Debye–Hückel approximation can't be used at the condition of high zeta potential and the exact solution of potential in cylindrical coordinates can't be obtained. Therefore, the Matlab bvp4c solver method and the finite difference method are employed to numerically solve the nonlinear Poisson–Boltzmann equation and the governing equations of the velocity distribution, respectively. To verify the validity of our numerical approach, a comparison has been made with the previous work in the case of low zeta potential and the excellent agreement between the solutions is clear. Then, in view of the obtained numerical solution for the velocity distribution, the numerical solutions of the flow rate and the shear stress are derived. Furthermore, based on numerical analysis, the influence of pertinent parameters on the potential distribution and the generation of flow is presented graphically.     

Visualizing the transient electroosmotic flow and measuring the zeta potential of microchannels with a micro-PIV technique

We have demonstrated a transient micro particle image velocimetry (micro-PIV) technique to measure the temporal development of electroosmotic flow in microchannels. Synchronization of different trigger signals for the laser, the CCD camera, and the high-voltage switch makes this measurement possible with a conventional micro-PIV setup. Using the transient micro-PIV technique, we have further proposed a method on the basis of inertial decoupling between the particle electrophoretic motion and the fluid electroosmotic flow to determine the electrophoretic component in the particle velocity and the zeta potential of the channel wall. It is shown that using the measured zeta potentials, the theoretical predictions agree well with the transient response of the electroosmotic velocities measured in this work.     

Theoretical analysis of electrokinetic flow and heat transfer in a microchannel under asymmetric boundary conditions

The main theme of the present work is to investigate the electrokinetic effects on liquid flow and heat transfer in a flat microchannel of two parallel plates under asymmetric boundary conditions including wall-sliding motion, unequal zeta potentials, and unequal heat fluxes on two walls. Based on the Debye-Huckel approximation, an electrical potential solution to the linearized Poisson-Boltzmann equation is obtained and employed in the analysis. The analytic solutions of the electrical potential, velocity distributions, streaming potential, friction coefficient, temperature distribution, and heat transfer rate are obtained, and thereby the effects of electrokinetic separation distance (K), zeta-potential level (zeta;(1)), ratio of two zeta potentials (r(zeta) identical with zeta;(2)/zeta;(1)), wall-sliding velocity (u(w)), and heat flux ratio (r(q) identical with q"(2)/q"(1)) are investigated. The present results reveal the effects of wall-sliding and zeta-potential ratio on the hydrodynamic nature of microchannel flow, and they are used to provide physical interpretations for the resultant electrokinetic effects and the underlying electro-hydrodynamic interaction mechanisms. In the final part the results of potential and velocity fields are applied in solving the energy equation. The temperature distributions and heat transfer characteristics under the asymmetrical kinematic, electric, and thermal boundary conditions considered presently are dealt with.     

Electrokinetic-vortex formation near a two-part cylinder with same-sign zeta potentials in a straight microchannel

Vortex formation near a two-part cylinder with zeta potentials of different values but the same sign under an external DC electric field is numerically investigated in this paper. The cylinder, inserted in a straight microchannel filled with an aqueous solution, is composed of an upstream part and a downstream part. When a DC electric field is applied in the channel, under certain conditions, the vortex will form near the cylinder due to the different velocities of electroosmotic flow generated on the cylinder surface. The numerical results reveal that the larger the velocity difference of electroosmotic flow generated on the two-part cylinder and the smaller the channel width, the more conducive to vortex formation in the channel. In addition, if the zeta potential ratios of cylinder downstream part to upstream part and channel wall to cylinder upstream part are unchanged, the DC electric field strength and the zeta potential value do not affect the pattern of vortices formed in the channel. This study provides a way for vortex formation in microchannels and has the potential application in microfluidic devices.     

Electrokinetic transport through nanochannels

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

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.     

Numerical Analysis of 3‐D Flow Past a Stationary Sphere with Slip Condition at Low and Moderate Reynolds Numbers

Computations are performed to determine the steady 3‐D viscous fluid flow forces acting on the stationary spherical suspended particle at low and moderate Reynolds numbers in the range of 0.1≤Re≤200. A slip is supposed on the boundary so that the slip velocity becomes proportional to the shear stress. This model possesses a single parameter to account for the slip coefficient λ (Pa.s/m), which is made dimensionless and is called Trostel number (Tr=λ a/μ). Decreasing slip, increases drag in all Reynolds limits, but slip has smaller effects on drag coefficient at lower Reynolds number regimes. Increasing slip at known Reynolds number causes to delay of flow separation and inflect point creation in velocity profiles. At full slip conditions, shear drag coefficient will be zero and radial drag coefficient reaches to its maximum values. Flow around of sphere at full‐slip condition is not equal to potential flow around a sphere. Present numerical results corresponding to full slip (Tr→0) are in complete accord with certain results of flow around of inviscid bubbles, and the results corresponding to no‐slip (Tr→∞) have excellent agreement with the results predicted by the no‐slip boundary condition.     

Lattice Poisson-Boltzmann simulations of electro-osmotic flows in microchannels

This paper presents the numerical results of electro-osmotic flows in micro- and nanofluidics using a lattice Poisson-Boltzmann method (LPBM) which combines a potential evolution method on discrete lattices to solve the nonlinear Poisson equation (lattice Poisson method) with a density evolution method on discrete lattices to solve the Boltzmann-BGK equation (lattice Boltzmann method). In an electrically driven osmotic flow field, the flow velocity increases with both the external electrical field strength and the surface zeta potential for flows in a homogeneous channel. However, for a given electrical field strength and zeta potential, electrically driven flows have an optimal ionic concentration and an optimum width that maximize the flow velocity. For pressure-driven flows, the electro-viscosity effect increases with the surface zeta potential, but has an ionic concentration that yields the largest electro-viscosity effect. The zeta potential arrangement has little effect on the electro-viscosity for heterogeneous channels. For flows driven by both an electrical force and a pressure gradient, various zeta potential arrangements were considered for maximize the mixing enhancement with a less energy dissipation.     

Effect of liquid slip in electrokinetic parallel-plate microchannel flow

Liquid slip at hydrophobic surfaces in microchannels has frequently been observed. We present here an analytical solution for oscillating flow in parallel-plate microchannels by combining the electrokinetic transport phenomena with Navier's slip condition. Our parametric results suggest that electrokinetic transport phenomena and liquid slip at channel walls are both important and should be considered simultaneously. Their significance depends on channel wall material, electrolyte concentration, and pH. For pressure-driven-flow, liquid slip counteracts the effect by the electrical double layer and induces a larger flow rate. A higher apparent viscosity would be predicted if slip is neglected. For electroosmotic flow, liquid slip alters the flow rate by about 20% for a thick electrical double layer. Our results provide design guidelines to precisely control time-dependent microflow in hydrophobic microfluidic microelectromechanical system devices.     

A method for simultaneously determining the zeta potentials of the channel surface and the tracer particles using microparticle image velocimetry technique

The zeta potentials of channel surfaces and tracer particles are of importance to the design of electrokinetic microfluidic devices, the characterization of channel materials, and the quantification of the microparticle image velocimetry (microPIV) measurement of EOFs. A method is proposed to simultaneously measure the zeta potentials of the channel surface and the tracer particles in aqueous solutions using the microPIV technique. Through the measurement of the steady velocity distributions of the tracer particles in both open- and closed-end rectangular microchannels under the same water chemistry condition, the electrophoretic velocity of the tracer particles and the EOF field of the microchannel are determined using the expressions derived in this study for the velocity distributions of charged tracer particles in the open- and closed-end rectangular microchannels. Thus, the zeta potentials of the tracer particles and the channel surfaces are simultaneously obtained using the least-square method to fit the microPIV measured velocity distribution of the tracer particles. Measurements were carried out with a microPIV system to determine the zeta potentials of the channel wall and the fluorescent tracer particles in deionized water and sodium chloride and boric acid solutions of various concentrations.     

Preparation, Properties and Mechanism of Inhomogeneous Calcium Alginate Ion Cross-linking Gel Microspheres

Inhomogeneous calcium alginate ion cross-linking gel microspheres,a novel ion absorbent,were prepared by dropping a sodium alginate solution to a calcium chloride solutioin via an electronic droplet generator.Calcium alginate microspheres have uniform particle sizes.a smooth surface and a microporous structure.The electrode probe reveals the inhomogeneous distribution of calcium ions with the highest concentration on the surface,and the lowest concentration in the cores of the spheres.As a novel ion adsorbent,calcium alginate gel microspheres have a lower limiting adsorption mass concentration,a higher enrichment capacity and a higher adsorption capacity for Pb^2 than usual ion exchange resins.The highest percentage of the adsorption is 99.79%.The limiting adsorption mass concentration is 0.0426mg/L.The adsorption capacity for Pb^2 is 644mg/g,Calcium alginate gel microspheres have a much faster ion exchange velocity than D418 chelating resin and D113 polyacrylate resin.The moving boundary model was employed to interpret the ion exchange kinetics process,which indicates that the ion exchange process is controlled by intraparticle diffusion of adsorbable ions.So the formation of inhomogeneous gel microspheres reduces the diffusion distance of adsorbable ions within the spheres and enhances the ion exchange velocity.Alginate has a higher selectivity for pb^2 than for Ca^2 and the selectivity coefficient KCa^Pb is 316. As an ion cross-linking gel,calcium alginate inhomogeneous microspheres can effectively adsorb heavy metal Pb^2 at a higher selectivity and a higher adsorption velocity.It is a novel and good ion adsorbent.     

Analysis of the slip effect on the permeate flux in membrane ultrafiltration

A method for predicting the mass transfer coefficient as well as the limiting permeate flux in membrane ultrafiltration has been found, based upon the boundary-layer theory which takes into account the slip velocity on the membrane surface. The theory presupposes the existence of a slip flow on a porous membrane surface, especially for the limiting permeate-flux operations. Further, the slip velocity increases with the size of the pores of the membrane, with feed velocity and also with feed concentration. The theory also showed that the permeate flux increases with the increase of the slip velocity. A considerable improvement in theoretical prediction of the permeate flux is expected if the slip flow effect is taken into consideration.     

Slip length of water on graphene: limitations of non-equilibrium molecular dynamics simulations

Data for the flow rate of water in carbon nanopores is widely scattered, both in experiments and simulations. In this work, we aim at precisely quantifying the characteristic large slip length and flow rate of water flowing in a planar graphene nanochannel. First, we quantify the slip length using the intrinsic interfacial friction coefficient between water and graphene, which is found from equilibrium molecular dynamics (EMD) simulations. We then calculate the flow rate and the slip length from the streaming velocity profiles obtained using non-equilibrium molecular dynamics (NEMD) simulations and compare with the predictions from the EMD simulations. The slip length calculated from NEMD simulations is found to be extremely sensitive to the curvature of the velocity profile and it possesses large statistical errors. We therefore pose the question: Can a micrometer range slip length be reliably determined using velocity profiles obtained from NEMD simulations? Our answer is "not practical, if not impossible" based on the analysis given as the results. In the case of high slip systems such as water in carbon nanochannels, the EMD method results are more reliable, accurate, and computationally more efficient compared to the direct NEMD method for predicting the nanofluidic flow rate and hydrodynamic boundary condition.