Electroosmotic Flow in Microchannels

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.     

Numerical modeling of Joule heating-induced temperature gradient focusing in microfluidic channels

Temperature gradient focusing (TGF) is a recently developed technique for spatially focusing and separating ionic analytes in microchannels. The temperature gradient required for TGF can be generated either by an imposed temperature gradient or by Joule heating resulting from an applied electric field that also drives the flow. In this study, a comprehensive numerical model describing the Joule heating induced temperature development and TGF is developed. The model consists of a set of governing equations including the Poisson-Boltzmann equation, the Laplace equation, the Navier-Stokes equations, the energy equations and the mass transport equation. As the thermophysical and electrical properties including the liquid dielectric constant, viscosity, and electric conductivity are temperature-dependent, these governing equations are coupled, and therefore the coupled governing equations are solved numerically by using a CFD-based numerical method. The numerical simulations agree well with the experimental results, suggesting the valid mathematical model presented in this study.     

Low-frequency velocity correlation spectrum of fluid in a rectangular microcapillary

In addition to the fast correlation for local stochastic motion, the velocity correlation function in a fluid enclosed within the pore boundaries features a slow long time-tail decay. At late times, the flow approaches that of an incompressible fluid. Here, we consider the motion of a viscous fluid, at constant temperature, in a rectangular semipermeable channel. The fluid is driven through the rectangular capillary by a uniform main pressure gradient. Tiny pressure gradients are allowed perpendicular to the main flux. We solve numerically the three-dimensional Navier-Stokes equations for the velocity field to obtain the steady solution. We then set and solve the Langevin equation for the fluid velocity. We report hydrodynamic fluctuations for the center-line velocity together with the corresponding relaxation times as a function of the size of the observing region and the Reynolds number. The effective diffusion coefficient for the fluid in the microchannel is also estimated (Deff = 1.43 x 10(-10) m2.s-1 for Re = 2), which is in accordance with measurements reported for a similar system (Stepisnik, J.; Callaghan, P. T. Physica B 2000, 292, 296-301; Stepisnik, J.; Callaghan, P. T. Magn. Reson. Imaging 2001, 19, 469-472).     

Influence of the Dukhin and Reynolds numbers on the apparent zeta potential of granular porous media

The Helmholtz-Smoluchowski (HS) equation is widely used to determine the apparent zeta potential of porous materials using the streaming potential method. We present a model able to correct this apparent zeta potential of granular media of the influence of the Dukhin and Reynolds numbers. The Dukhin number represents the ratio between the surface conductivity (mainly occurring in the Stern layer) and the pore water conductivity. The Reynolds number represents the ratio between inertial and viscous forces in the Navier-Stokes equation. We show here that the HS equation can lead to serious errors if it is used to predict the dependence of zeta potential on flow in the inertial laminar flow regime without taking into account these corrections. For indifferent 1:1 electrolytes (such as sodium chloride), we derived two simple scaling laws for the dependence of the streaming potential coupling coefficient (or the apparent zeta potential) on the Dukhin and Reynolds numbers. Our model is compared with a new set of experimental data obtained on glass bead packs saturated with NaCl solutions at different salinities and pH. We find fairly good agreement between the model and these experimental data.     

Numerical simulation of bubble dynamics in a micro-channel under a nonuniform electric field

A numerical method is used to simulate the motion and coalescence of air bubbles in a micro-channel under a nonuniform electric field. The channel is equipped with arrays of electrodes embedded in its wall and voltages are applied on the electrodes to generate a specified electric field gradient in the longitudinal direction. In the study, the Navier-Stokes equations are solved by using the level set method handling the deformable/moving interfaces between the bubbles and the ambient liquid. Both the polarization Coulomb force and the dielectrophoresis force are considered as the force source of the Navier-Stokes equations by solving the Maxwell's equations. The flow field equations and the electric field equations are coupled and solved by using the finite element method. The electric field characteristics and the dynamic behavior of a bubble are analyzed by studying the distributions of the electric field and the force, the deformation and the moving velocity of the air bubble. The result suggests that the model of dispersed drops suspended in the immiscible dielectric liquid and driven by a nonuniform electric field is an effective method for the transportation and coalescence of micro-drops.     

Electrokinetic translocation of a deformable nanoparticle controlled by field effect in nanopores

Nanopores have become a popular single-molecule manipulation and detection technology. In this paper, we have constructed a continuum model of the nanopore; the arbitrary Lagrangian-Eulerian (ALE) method is used to describe the motion of particles and fluid. The mathematical model couples the stress-strain equation for the dynamics of a deformable particle, the Poisson equation for the electric field, the Navier-Stokes equations for the flow field, and the Nernst-Planck equations for ionic transport. Based on the model, the mechanism of field-effect regulation of particles passing through a nanopore is investigated. The results show that the transport of particles which is controlled by the field effect depends on the electroosmotic flow (EOF) generated by the gate electrode in the nanopore and the electrostatic interaction between the nanopore and particles. That also explains the asymmetry of particle transport velocity in the nanopore with a gate electrode. When the gate potential is negative, or the gate electrode length is small, the maximum deformation of the particles is increased. The field-effect regulation in the nanopore provides an active and compatible method for nanopore detection, and provides a convenient method for the active control of the particle deformation in the nanopore.     

Microdroplet deposition under a liquid medium

An experimental and numerical study of the factors affecting the reproducibility of microdroplet depositions performed under a liquid medium is presented. In the deposition procedure, sample solution is dispensed from the end of a capillary by the aid of a pressure pulse onto a substrate with pillar-shaped sample anchors. The deposition was modeled using the convective Cahn-Hilliard equation coupled with the Navier-Stokes equations with added surface tension and gravity forces. To avoid a severe time-step restriction imposed by the fourth-order Cahn-Hilliard equation, a semi-implicit scheme was developed. An axisymmetric model was used, and an adaptive finite element method was implemented. In both the experimental and numerical study it was shown that the deposited volume mainly depends on the capillary-substrate distance and the anchor surface wettability. A critical equilibrium contact angle has been identified below which reproducible depositions are facilitated.     

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.     

Diffusioosmotic flows in slit nanochannels

Diffusioosmotic flows of electrolyte solutions in slit nanochannels with homogeneous surface charges induced by electrolyte concentration gradients in the absence of externally applied pressure gradients and potential differences are investigated theoretically. A continuum mathematical model consisting of the strongly coupled Nernst-Planck equations for the ionic species' concentrations, the Poisson equation for the electric potential in the electrolyte solution, and the Navier-Stokes equations for the flow field is numerically solved simultaneously. The induced diffusioosmotic flow through the nanochannel is computed as functions of the externally imposed concentration gradient, the concentration of the electrolyte solution, and the surface charge density along the walls of the nanochannel. With the externally applied electrolyte concentration gradient, a strongly spatially dependent electric field and pressure gradient are induced within the nanochannel that, in turn, generate a spatially dependent diffusioosmotic flow. The diffusioosmotic flow is opposite to the applied concentration gradient for a relatively low bulk electrolyte concentration. However, the electrolyte solution flows from one end of the nanochannel with a higher electrolyte concentration to the other end with a lower electrolyte concentration when the bulk electrolyte concentration is relatively high. There is an optimal concentration gradient under which the flow rate attains the maximum. The induced flow is enhanced with the increase in the fixed surface charge along the wall of the nanochannel for a relatively low bulk electrolyte concentration.     

Statistical theory of turbulent mass transfer in electrochemical systems

The paper presents an overview of the statistical theory of turbulent mass transfer in electrochemical systems and some new results which generalize the previously obtained relations for the flows of complex geometry. The developed theory does not use traditional semi-empirical hypotheses and analogies, but directly addresses to the solving of the critical for turbulent transfer the closure problem. The mathematical procedure for solving of the closure problem makes use of new equations for the correlations between concentration and velocity fluctuations in different space points and at different time moments; the dumping of turbulent pulsations in the viscous sublayer allows to neglect high order moments and obtain a closed equation for the turbulent mass flux. In general, the relation between the turbulent mass flux and the mean concentration gradient is non-local. Using available experimental information, the non-local equation for the turbulent mass flux is reduced to the traditional local one and the functional form of the turbulent diffusion coefficient is obtained. It is demonstrated that Reynolds analogy cannot been used for the prediction of the turbulent diffusivity. Applications of the developed theory to chemical engineering and to electrochemical flow diagnostics (prediction of flow characteristics using limiting diffusion current measurements) are discussed.     

Sedimentation of spheres at small Reynolds number

The effect of fluid inertia on the settling of spheres in a viscous incompressible fluid is studied in the limit of small Reynolds number. The kinetic energy of flow depends on the positions of the spheres, and gives rise to forces on the spheres. In the dilute limit it suffices to study the corresponding pair interaction. The interaction is calculated from the Stokes flow for two spheres settling between plane walls in the point particle limit. The dissipative interaction between a pair of spheres is calculated from the Proudman-Pearson [I. Proudman and J. R. A. Pearson, J. Fluid Mech. 2, 237 (1957)] solution of the Navier-Stokes equations for flow about a sphere in unbounded geometry. The combination of kinetic and dissipative interaction gives rise to a repulsive force of range of the order of the sphere diameter divided by the Reynolds number.     

A finite element formulation of frequency-dependent electro-osmosis

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.     

Analysis of electroosmotic flow with step change in zeta potential

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.     

Numerical analysis of the thermal effect on electroosmotic flow and electrokinetic mass transport in microchannels

Joule heating is present in electrokinetically driven flow and mass transport in microfluidic systems. Nowadays, there is a trend of replacing costly glass-based microfluidic systems by the disposable, cheap polymer-based microfluidic systems. Due to poor thermal conductivity of polymer materials, the thermal management of the polymer-based microfluidic systems may become a problem. In this study, numerical analysis is presented for transient temperature development due to Joule heating and its effect on the electroosmotic flow (EOF) and mass species transport in microchannels. The proposed model includes the coupling Poisson-Boltzmann (P-B) equation, the modified Navier-Stokes (N-S) equations, the conjugate energy equation, and the mass species transport equation. The results show that the time development for both the electroosmotic flow field and the Joule heating induced temperature field are less than 1 s. The Joule heating induced temperature field is strongly dependent on channel size, electrolyte concentration, and applied electric field strength. The simulations reveal that the presence of the Joule heating can result in significantly different characteristics of the electroosmotic flow and electrokinetic mass transport in microchannels.     

On the spreading and solidification of molten particles in a plasma spray process effect of thermal contact resistance

The spreading and simultaneous solidification of a liquid droplet upon its impingement onto a substrate permitting thermal contact resistance has been numerically simulated; the effect of contact resistance and the importance of solidification on droplet spreading are investigated. The numerical solution for the complete Navier-Stokes equations is based on the modified SOLA-VOF method using rectangular mesh in axisymmetric geometry. The solidification of the deforming droplet is considered by a one-dimensional heat conduction model. The predictions are in good agreement with the available experimental data and the model may be well suited for investigating droplet impact and simultaneous solidification permitting contact resistance at the substrate. We found that the final splat diameter could be extremely sensitive to the magnitude of the thermal contact resistance. The results also show that for the condition of higher Reynolds and/or higher Stefan numbers the effect of solidification on the final splat diameter is more important.     

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.     

Modeling electroosmotic and pressure-driven flows in porous microfluidic devices: zeta potential and porosity changes near the channel walls

This work presents analytical solutions for both pressure-driven and electroosmotic flows in microchannels incorporating porous media. Solutions are based on a volume-averaged flow model using a scaling of the Navier-Stokes equations for fluid flow. The general model allows analysis of fluid flow in channels with porous regions bordering open regions and includes viscous forces, permitting consideration of porosity and zeta potential variations near channel walls. To obtain analytical solutions problems are constrained to the linearized Poisson-Boltzmann equation and a variation of Brinkman's equation [Appl. Sci. Res., Sect. A 1, 27 (1947); 1, 81 (1947)]. Cases include one continuous porous medium, two adjacent regions of different porosities, or one open channel adjacent to a porous region, and the porous material may have a different zeta potential than that of the channel walls. Solutions are described for two geometries, including flow between two parallel plates or in a cylinder. The model illustrates the relative importance of porosity and zeta potential in different regions of each channel.     

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.