Roles of solution conductivity mismatch in transient current and fluid transport in electrolyte displacement by electro-osmotic flow

Electro-osmotically driven displacement between two solutions having a conductivity mismatch is theoretically examined. Internal pressures induced by the conductivity mismatch can affect the propagation of the solution interface and the behavior of the transient current. Combining Ohm’s law and fluid mass conservation, we derive a coupled set of length-averaged equations accounting for how the electric current and the traveling distance of the solution interface vary with time, electric field, and the solution conductivities. Extension to successive displacements involving multiple solution zones is made to reveal non-monotonic and stagewise changes in transient currents. For the first time, critical roles of surface conductance on displacements in highly charged channels are unraveled. We show that if the lower conductivity solution has a greater valence than the higher one, the effective conductivity of the former can exceed that of the latter when the channel height is below some critical value. The resulting transient current behavior can turn opposite to that usually observed in the large-channel case, offering a new paradigm for gauging the importance of surface conductance in submicron charged channels. Possible impacts of diffusion smearing and hydrodynamic dispersion are also discussed by including the additional mixing zone into the analysis. Having shown good agreement with the existing experimental data, our analysis not only captures the natures of solution displacement by electro-osmotic flow (EOF), but also extends the applicability of the current monitoring method for measuring surface zeta potentials of microchannels.     

Electrokinetic motion of an electrically induced Janus droplet in microchannels

The electrokinetic motion of an electrically induced Janus oil droplet with one side covered with an aluminum oxide nanoparticle film in a circular microchannel was numerically simulated in this paper. The Janus oil droplet is electrically anisotropic as the nanoparticle-covered area carries positive charges and the rest oil–water surface area carries negative charges. A theoretical model was constructed to calculate the electrokinetic velocity of the Janus droplet by considering the force balance on the surface of the Janus droplet at steady state. In the model, the effects of the electric double layer and surface charges on the motion at the oil–water interface are considered. The effects of five parameters on the electrokinetic motion of the Janus droplets were studied: the electric field, the zeta potential ratio of the positively charged side to the negatively charged side of the Janus droplet, the viscosity ratio of the oil phase to the water phase, the nanoparticle coverage of the Janus droplet, and the size ratio of the diameter of the Janus droplet to the diameter of the cylindrical microchannel. The simulation results indicate that the increase in the electrical field, the zeta potential ratio, the viscosity ratio or the nanoparticle coverage leads to faster electrokinetic motion of the Janus droplet. On the other hand, with the increase in size ratio, the electrokinetic velocity of Janus droplet first decreases gradually then increases sharply. The simulated results were compared with the experimental results and good agreement was found.     

Electrophoresis and diffusiophoresis of a colloidal sphere with double-layer polarization in a concentric charged cavity

The electrophoretic and diffusiophoretic motions of a dielectric spherical particle situated at the center of a spherical cavity filled with an electrolyte solution are studied analytically. The applied electric field and electrolyte concentration gradient are uniform; the electric double layers at the particle surface and cavity wall are thin relative to the radius of the particle and distance between the solid surfaces, but the diffuse-layer polarization effect over the particle surface is considered. After solving the equations of conservation governing the systems, explicit formulas for the electrophoretic and diffusiophoretic velocities of the confined particle are obtained and their results relative to those of a particle under identical conditions in an unbounded solution are presented for various values of the radius ratio and zeta potential ratio between the particle and the cavity and of other parameters in the systems. The contributions from the electroosmotic and diffusioosmotic flows occurring along the cavity wall and from the wall-corrected electrophoretic and diffusiophoretic driving forces to the particle velocities are equivalently important and can be superimposed due to the linearity of the problems. The normalized migration velocities of the particle in general decrease with an increase in the particle-to-cavity radius ratio and increase with an increase in the cavity-to-particle zeta potential ratio. The effects of the charged cavity wall on the electrokinetic migrations of the particle are significant and can reverse their directions.     

Diffusioosmosis of electrolyte solutions in fibrous porous media

An analytical study is presented for the diffusioosmotic flow of an electrolyte solution in the fibrous medium constructed by an ordered array of parallel charged circular cylinders at the steady state. The prescribed electrolyte concentration gradient is constant but can be oriented arbitrarily with respect to the axes of the cylinders. The electric double layer surrounding each cylinder may have an arbitrary thickness relative to the radius of the cylinder. A unit cell model which allows for the overlap of the double layers of adjacent cylinders is employed to account for the effect of fibers on each other. The electrostatic potential distribution in the fluid phase of a cell is obtained by solving the linearized Poisson–Boltzmann equation, which applies to the case of low surface potential of the cylinders. The macroscopic electric field induced by the imposed electrolyte concentration gradient through the fluid phase in a cell is determined as a function of the radial position. A closed-form formula for the fluid velocity profile of the electrolyte solution due to the combination of electroosmotic and chemiosmotic contributions as a function of the porosity of the array of cylinders correct to the second order of their surface charge density or zeta potential is derived as the solution of a modified Navier–Stokes equation. The diffusioosmotic velocity can have more than one reversal in direction over a small range of the zeta potential. For a given electrolyte concentration gradient in a cell, the fluid flow rate does not necessarily increase with an increase in the electrokinetic radius of the cylinder, which is the cylinder radius divided by the Debye screening length. The effect of the radial distribution of the induced axial electric field in the double layer on the diffusioosmotic flow is found to be of dominant significance in most practical situations.     

A Coupled Lattice Boltzmann Method to Solve Nernst–Planck Model for Simulating Electro-osmotic Flows

In this paper, we focus on the nonlinear coupling mechanism of the Nernst–Planck model and propose a coupled lattice Boltzmann method (LBM) to solve it. In this method, a new LBM for the Nernst–Planck equation is developed, a multi-relaxation-time (MRT)-LBM for flow field and an LBM for the Poisson equation are used. And then, we discuss the choice of the model and found that the MRT-LBM is much more stable and accurate than the LBGK model. A reasonable iterative sequence and evolution number for each LBM are proposed by considering the properties of the coupled LBM. The accuracy and stability of the presented coupled LBM are also discussed through simulating electro-osmotic flows (EOF) in micro-channels. Furthermore, to test the applicability of it, the EOF with non-uniform surface potential in micro-channels based on the Nernst–Planck model is simulated. And we investigate the effects of non-uniform surface potential on the pattern of the EOF at different external applied electric fields. Finally, a comparison of the difference between the Nernst–Planck model and the Poisson–Boltzmann model is presented.     

A mathematical model for a class of frying processes

We present a mathematical model for deep frying in absence of mechanical deformation and in one-dimensional geometry. In the generic stage of the process the inner zone is saturated with liquid water below the boiling point at atmospheric pressure. When the boiling point is reached partial vaporization occurs and a zone of water–vapor thermodynamical equilibrium is formed, followed by a region of pure vapor. The interface between mixed and vapor region can be either a zero saturation surface or a level set of pressure when the latter reaches an imposed constraint. The outmost layer is crust. In each region the governing equations are written for temperature and pressure and the conditions at the interfaces comes out as the corresponding Rankine–Hugoniot relations. Boundary conditions at the crust–oil interface are discussed. Rescaling leads to only moderate simplifications and we are left with a considerably difficult free boundary problem for a parabolic system.     

Pressure-driven electrokinetic slip flows of viscoelastic fluids in hydrophobic microchannels

This work investigates the steady-state slip flow of viscoelastic fluids in hydrophobic two-dimensional microchannels under the combined influence of electro-osmotic and pressure gradient forcings with symmetric or asymmetric zeta potentials at the walls. The Debye–Hückel approximation for weak potential is assumed, and the simplified Phan-Thien-Tanner model was used for the constitutive equation. Due to the different hydrophobic characteristics of the microchannel walls, we study the influence of the Navier slip boundary condition on the fluid flow, by considering different slip coefficients at both walls and varying the electrical double-layer thickness, the ratio between the applied streamwise gradients of electric potential and pressure, and the ratio of the zeta potentials. For the symmetric case, the effect of the nonlinear Navier slip model on the fluid flow is also investigated.     

Diffusioosmotic flow of electrolyte solutions in fibrous porous media at arbitrary zeta potential and double-layer thickness

The diffusioosmosis of electrolyte solutions in the fibrous medium constructed by a homogeneous assemblage of parallel, charged, circular cylinders caused by a uniform concentration gradient prescribed in their axial direction is studied theoretically. The electric double layers adjacent to the cylinders may have an arbitrary thickness. A unit cell model is used to account for the interaction effect among the cylinders. The electrostatic potential distribution in the fluid phase is determined with an analytical approximation to the solution of the Poisson–Boltzmann equation. By solving the fluid momentum equation with the constraint of no net electric current arising from the co-current diffusion, electric migration, and diffusioosmotic convection of the electrolyte ions, the macroscopic electric field and the fluid velocity in the axial direction induced by the applied electrolyte concentration gradient are obtained semi-analytically as functions of the radial position in a cell in a self-consistent manner. The magnitude and direction of the diffusioosmotic flow relative to the concentration gradient are determined by the combination of the porosity of the array of cylinders, the zeta potential of the cylinders, the properties of the electrolyte solution, and other relevant factors. The fluid velocity generally increases with increasing porosity of the array of cylinders, but there are exceptions. The effects of the radial distribution of the induced electric field and of the ionic convection in the double layers on the diffusioosmotic flow are significant.     

Dispersion in electroosmotic flow generated by oscillatory electric field interacting with oscillatory wall potentials

An analytical study is presented in this article on the dispersion of a neutral solute released in an oscillatory electroosmotic flow (EOF) through a two-dimensional microchannel. The flow is driven by the nonlinear interaction between oscillatory axial electric field and oscillatory wall potentials. These fields have the same oscillation frequency, but with disparate phases. An asymptotic method of averaging is employed to derive the analytical expressions for the steady-flow-induced and oscillatory-flow-induced components of the dispersion coefficient. Dispersion coefficients are functions of various parameters representing the effects of electric double-layer thickness (Debye length), oscillation parameter, and phases of the oscillating fields. The time–harmonic interaction between the wall potentials and electric field generates steady as well as time-oscillatory components of electroosmotic flow, each of which will contribute to a steady component of the dispersion coefficient. It is found that, for a thin electric double layer, the phases of the oscillating wall potentials will play an important role in determining the magnitude of the dispersion coefficient. When both phases are zero (i.e., full synchronization of the wall potentials with the electric field), the flow is nearly a plug flow leading to very small dispersion. When one phase is zero and the other phase is π,?the flow will be sheared to the largest possible extent at the center of the channel, and such a sharp velocity gradient will lead to the maximum possible dispersion coefficient.     

Lattice Boltzmann method simulation of electroosmotic stirring in a microscale cavity

The suitable surface modification of microfluidic channels can enable a neutral electrolyte solution to develop an electric double layer (EDL). The ions contained within the EDL can be moved by applying an external electric field, inducing electroosmotic flows (EOFs) that results in associated stirring. This provides a solution for the rapid mixing required for many microfluidic applications. We have investigated EOFs generated by applying a steady electric field across a square cavity that has homogenous electric potentials along its walls. The flowfield is simulated using the lattice Boltzmann method. The extent of mixing is characterized for different electrode configurations and electric field strengths. We find that rapid mixing can be achieved by using this simple configuration which increases with increasing electric field strength. The mixing time for water-soluble organic molecules can be decreased by four orders of magnitude by suitable choice of wall zeta potential and electric field. We dedicate this paper to the memory of our colleagues Professors Kevin Granata and Liviu Librescu who fell tragically on April 16, 2007 while answering their call to serve higher education. They continue to inspire us. AM gratefully acknowledges support from Jadavpur University under the World Bank funded Technical Education Quality Improvement Programme of the Government of India and the hospitality of the Virginia Tech ESM Department where he conducted a portion of this work.     

Diffusiophoresis of a charged particle in a charged cavity with arbitrary electric double layer thickness

An analysis is presented for the diffusiophoretic motion of a charged colloidal sphere located at the center of a charged spherical cavity filled with an electrolyte solution at the quasisteady state for the case of arbitrary electric double layers. The electrokinetic equations governing the ionic concentration, electric potential, and velocity distributions in the fluid are linearized with assumption that the system is slightly distorted from equilibrium. These linearized differential equations are solved using a perturbation method with the zeta potentials of the particle and cavity as the small perturbation parameters. An explicit formula for the diffusiophoretic velocity of the particle as a combination of the electrophoretic and chemiphoretic contributions valid for arbitrary values of \(\kappa a\) and \(a/b\) is obtained by balancing the electrostatic and hydrodynamic forces exerted on it, where \(\kappa\) is the Debye screening parameter, \(a\) is the radius of the particle, and \(b\) is the radius of the cavity. The effect of the charged cavity wall on the diffusiophoresis of the particle is interesting and can be significant. The contributions from the diffusioosmotic (electroosmotic and chemiosmotic) flow taking place along the cavity wall and from the wall-corrected diffusiophoretic force to the particle velocity are comparably important, and this diffusioosmotic flow can reverse the direction of diffusiophoresis. The particle velocity in general increases with an increase in \(\kappa a\) and decreases with an increase in \(a/b\), but exceptions exist.     

Fully explicit dissipative particle dynamics simulation of electroosmotic flow in nanochannels

A fully explicit mesoscale simulation of electroosmotic flow (EOF) in nanochannels is presented by an extended dissipative particle dynamics (DPD) method. To avoid formation of ionic pairs through interacting soft-core charges, a Slater-type smearing distribution borrowed from quantum mechanics is utilized to surround each soft DPD ion with a charge cloud. To account for reduced periodicity normal to the walls direction, a corrected version of 3D Ewald sum is implemented in which a dipole moment term is deducted from energy and force terms of non-frozen charges. Simulation box is then elongated normal to walls to dampen spurious interslab interactions by adding vacuum gaps between periodic images. These measures together with the established unit conversions guarantee perfect match to molecular dynamics results. The transition of EOF velocity profile from parabolic (equivalent to overlap of electric double layers) to plug-like shapes is studied across the changing electric field between 0.06 and 0.41 [V/nm], and varying salt concentration from 0.26 to 2.0 [M]. It is found that 1.25 [V] increase in the driving voltage can potentially enhance the electroosmotic flow rate by 8–11 times in the range of ionic concentrations studied. The range of surface zeta potential calculated as \( 27 < \zeta < 52 \) [mV] in the linear response regime, as identified to occur for 0.24 ≤ E [V/nm], agrees reasonably with numerical and experimental studies.     

Electrophoresis of a colloidal sphere with double-layer polarization in a microtube

A theoretical study is presented for the electrophoretic motion of a spherical particle in an electrolyte solution along the axis of a circular microtube, whose wall may be either insulating or prescribed with the linear far-field electric potential distribution. The electric double layers adjoining the charged particle surface and tube wall are finitely thin, and the polarization of the diffuse layer at the particle surface is allowed. The general solutions to the electrostatic and hydrodynamic governing equations are constructed in combined cylindrical and spherical coordinates, and the boundary conditions are enforced on the tube wall by the Fourier transform and along the particle surface by a collocation method. The collocation results for the electrophoretic mobility of the confined particle, which agree well with the asymptotic formulas obtained by using a method of reflections, are obtained for various values of the particle, wall, and solution characteristics. An insulating tube wall and a tube wall with the far-field potential distribution affect the electrophoresis of the particle quite differently. Although the particle mobility in a tube with uncharged wall in general decreases with an increase in the particle-to-tube radius ratio a/b, it can increase with an increase in a/b as this ratio is close to unity for some cases because of the competition between the wall effects of hydrodynamic retardation and possible electrochemical enhancement on the particle migration. When the zeta potential of the tube wall is comparable to that of the particle, the electroosmotic flow of the bulk fluid induced by the tube wall dominates the electrokinetic migration of the particle.     

Nonlinear surface waves interacting with a linear shear current

To describe the evolution of fully nonlinear surface gravity waves in a linear shear current, a closed system of exact evolution equations for the free surface elevation and the free surface velocity potential is derived using a conformal mapping technique. Traveling wave solutions of the system are obtained numerically and it is found that the maximum wave amplitude for a positive shear current is much smaller than that in the absence of any shear while the opposite is true for a negative shear current. The new evolution equations are also solved numerically using a pseudo-spectral method to study the Benjamin–Feir instability of a modulated wave train in both positive and negative shear currents. With a fixed wave slope, compared with the irrotational case, the envelope of the modulated wave train grows faster in a positive shear current and slower in a negative shear current.     

Controlling electroosmotic flow by polymer coating: a dissipative particle dynamics study

We have performed dissipative particle dynamics (DPDs) simulations of electroosmotic flow (EOF) through a polymer-grafted nanopore. In this model, charged particles including salt ions and counterions are not included explicitly, and EOF is created using an effective boundary condition. The screening effect of polymer layer on EOF is investigated in detail under different solvent qualities and boundary electroosmotic velocities. Results show that the solvent quality has a significant effect on the conformational properties of polymer chains and the flow characteristics of the solvent. The polymer layer undergoes a collapsed transition when decreasing the solvent quality from good to poor. Under different solvent qualities, enhancing the EOF leads to a different variation tendency of the layer thickness. The solvent-induced permeability change is inconsistent with the steady velocity away from the surface. The minimum value of the solvent permeability occurs at an intermediate solvent quality. However, the layer thickness drops gradually to a smallest value (corresponding to the largest effective pore radius) in the poor solvent condition. It is also found that the polymer inclination and stretching length exhibit a complex behavior under the combined effect of solvent quality and electroosmosis-induced shear.     

Hydrodynamic dispersion by electroosmotic flow of viscoelastic fluids within a slit microchannel

The biofluids being manipulated in lab-on-a-chip devices usually contain elastic macromolecules. Accordingly, for an accurate modeling of the relevant flow physics one should invoke viscoelastic constitutive equations. In this paper, attention is paid toward the hydrodynamic dispersion by the fully developed electroosmotic flow of PTT viscoelastic fluids in slit microchannels of low zeta potential. Adopting the Taylor–Aris approach, analytical solutions are derived for late-time solute concentration and effective dispersion coefficient. Finite element-based numerical simulations are also conducted to monitor the broadening of an analyte band from the moment of injection. Both approaches are found to be in a good agreement with an average error of below 8% in the calculated dispersion coefficients. It is observed that, for given zeta potential and electrolyte type, the hydrodynamic dispersion is severely pronounced by increasing the level of elasticity in the fluid. The variations are reversed and less pronounced when the analysis is made for a fixed mean flow rate. Moreover, the effective dispersivity grows by thickening the EDL when it is sufficiently thin, whereas the opposite is true for thick EDLs. Finally, an inspection of the average concentration reveals the formation of tails for thick EDLs that may reduce the resolution in sensing applications.     

Electrophoretic motion of two spherical particles in a rectangular microchannel

The electrophoretic motion of two spherical particles in an aqueous electrolyte solution in a small rectangular microchannel was studied in this paper. A theoretical model was developed to describe the electric field, the flow field, and the particle motion. A direct numerical simulation method using the finite element method is employed to solve the model. The simulation results clearly show how the presence of one particle influences the electric field, the flow field, and the motion of the adjacent particle. Such an influence weakens with the separation distance. In addition to the zeta potentials, the particles motion depends on their sizes: the smaller particle moves slightly faster. For a faster particle moving from behind of a slower particle, the simulation results show that the faster particle will climb and then pass the slower particle when the two particles centers are not located on the same line parallel to the applied electric field.     

Reviewing the mathematical validity of a fuel cell cathode model. Existence of weak bounded solution

We consider a system of nonlinear PDEs in a domain with a triple phase boundary, describing electrochemical processes in a mixed conduction, solid-oxide cathode of a fuel cell. It represents oxygen diffusion (with nonlinear diffusion coefficient) in the gas phase, oxygen ion diffusion in the bulk phase, electron diffusion in the electrolyte, surface exchange (nonlinear) on the interface of gas and the (mixed conduction) electrode material and finally charge transfer (nonlinear) at the interface between the electrolyte and the electrode material. We prove the validity of the model both mathematically and numerically. In fact, we prove the existence of a bounded weak solution using the Schauder fixed point theorem. We calculate the numerical solutions for given function and parameter values, and show that they correspond to theoretical results. In particular, we provide a numerical confirmation of the a priori bounds.     

Electric field-induced concentration and capture of DNA onto microtips

Simple, high-yield concentration of DNA is important for high-throughput genetic analysis and disease diagnosis. Glass-based microfilters are popular but the process requires centrifugation steps with cumbersome chemical processes. As an alternative, a concentration method using an electric field has been explored previously, but with limited efficiency. In this paper, electric field-induced concentration and capture of DNA are studied by using high-aspect-ratio microtips coated with a gold layer. The microtips are immersed longitudinally into a solution of 100???L containing ??-phage DNA. After DNA concentration using an electric field, the microtips are withdrawn from the solution. Under AC- and biased AC fields, DNA is concentrated by electrophoresis (EP), dielectrophoresis (DEP), and electroosmotic flow (EOF). To reduce capillary effects in the withdrawal process, the microtips are coated with positively charged poly-l-lysine (PLL). The pattern of captured DNA is analyzed by fluorescence microscopy. DEP attracts DNA molecules at the edges of microtips, where the highest gradient of electric field exists. EP attracts DNA onto the surface of microtips following the vectors of an electric field. EOF generates vortexes that deliver DNA onto microtips. Using this method, 85% of DNA is captured on the PLL-coated microtips after three sequential captures. The concentration mechanism can potentially facilitate rapid and simple preparation of DNA for downstream analysis.     

Selective particle and cell clustering at air–liquid interfaces within ultrasonic microfluidic systems

In this study, we demonstrate particle and cell clustering in distinct patterns on the free surface of microfluidic volumes. Employing ultrasonic actuation, submersed microparticles are forced to two principal positions: nodal lines (pressure minima) of a standing wave within the liquid bulk, and distinct locations on the air–liquid interface (free surface); the latter of which has not been previously demonstrated using ultrasonic standing waves. As such, we unravel the fundamental mechanisms behind such patterns, showing that the contribution of fluid particle velocity variations on the free surface (acoustic radiation force) results in patterned particle clustering. In addition, by varying the size and density of the microparticles (3.5–31 μm polystyrene and 1–5 μm silica), acoustic streaming is found to increase the tendency for a smaller and lighter particle to cluster at the air–liquid interface. This selectivity is exploited for the isolation of multiple microparticle and cell types on the free surface from their nodally aligned counterparts. Free surface clustering is demonstrated in both an open microfluidic chamber and a sessile droplet, as well as using a range of biological species Escherichia coli, blood cells, Ragweed pollen and Paper Mulberry pollen). The ability to selectively cluster submersed microparticles and cells in distinct patterns on the free surface showcases the excellent suitability of this method to lab-on-a-chip systems.