Diffusioosmosis of electrolyte solutions in a fine capillary slit

The steady diffusioosmotic flows of an electrolyte solution along a charged plane wall and in a capillary channel between two identical parallel charged plates generated by an imposed tangential concentration gradient are theoretically investigated. The plane walls may have either a constant surface potential or a constant surface charge density. The electrical double layers adjacent to the charged walls may have an arbitrary thickness and their electrostatic potential distributions are determined by the Poisson-Boltzmann equation. Solving a modified Navier-Stokes equation with the constraint of no net electric current arising from the cocurrent diffusion, electric migration, and diffusioosmotic convection of the electrolyte ions, the macroscopic electric field and the fluid velocity along the tangential direction induced by the imposed electrolyte concentration gradient are obtained semianalytically as a function of the lateral position in a self-consistent way. The direction of the diffusioosmotic flow relative to the concentration gradient is determined by the combination of the zeta potential (or surface charge density) of the wall, the properties of the electrolyte solution, and other relevant factors. For a given concentration gradient of an electrolyte along a plane wall, the magnitude of fluid velocity at a position in general increases with an increase in its electrokinetic distance from the wall, but there are exceptions. The effect of the lateral distribution of the induced tangential electric field and the relaxation effect in the double layer on the diffusioosmotic flow are found to be very significant.     

Electric conductivity in a fibrous porous medium with thin but polarized double layers

The electric conduction in the fibrous medium constructed by a homogeneous array of parallel, identical, charged, circular cylinders having an arbitrary zeta potential filled with the solution of a symmetrically charged electrolyte is analytically examined. The thickness of the electric double layers surrounding the dielectric cylinders is assumed to be small relative to the radius of each cylinder and to the gap width between two neighboring cylinders, but the polarization of the mobile ions in the diffuse layers is allowed. The effect of interactions among individual cylinders is taken into explicit account by employing a unit cell model. The appropriate equations of conservation of electrochemical potential energies of ionic species are solved for each cell, in which a cylinder is envisaged to be surrounded by a coaxial cylindrical shell of the fluid solution. Analytical expressions for the effective electric conductivity are obtained in closed forms as functions of the porosity of the fiber matrix and other characteristics of the porous system. Comparisons of the results of the cell model with different conditions at the outer boundary of the cell are made. Under an otherwise identical condition, the electric conductivity in a porous medium composed of an array of parallel cylinders in the transverse direction is smaller than that of a suspension of spheres. The effect of interactions among the cylinders or spheres on the effective conductivity can be quite significant under appropriate conditions.     

Diffusioosmosis of electrolyte solutions along a charged plane wall

The steady diffusioosmotic flow of an electrolyte solution along a dielectric plane wall caused by an imposed tangential concentration gradient is analytically examined. The plane wall may have either a constant surface potential or a constant surface charge density of an arbitrary quantity. The electric double layer adjacent to the charged wall may have an arbitrary thickness, and its electrostatic potential distribution is determined by the Poisson-Boltzmann equation. The macroscopic electric field along the tangential direction induced by the imposed electrolyte concentration gradient is obtained as a function of the lateral position. A closed-form formula for the fluid velocity profile is derived as the solution of a modified Navier-Stokes equation. The direction of the diffusioosmotic flow relative to the concentration gradient is determined by the combination of the zeta potential of the wall and the properties of the electrolyte solution. For a given concentration gradient of an electrolyte along a plane wall, the magnitude of fluid velocity at a position in general increases with an increase in its electrokinetic distance from the wall, but there are exceptions. The effect of the lateral distribution of the induced tangential electric field in the double layer on the diffusioosmotic flow is found to be very significant and cannot be ignored.     

Diffusioosmosis of electrolyte solutions in a capillary slit with adsorbed polyelectrolyte layers

A theoretical study is presented for the steady diffusioosmotic flow of an electrolyte solution in a fine capillary slit with each of its inside walls coated with a layer of polyelectrolytes generated by an imposed tangential concentration gradient. In this solvent-permeable and ion-penetrable surface charge layer, idealized polyelectrolyte segments are assumed to be distributed at a uniform density. The electric double layer and the surface charge layer may have arbitrary thicknesses relative to the gap width between the slit walls. The Poisson-Boltzmann equation and a modified Navier-Stokes/Brinkman equation are solved numerically to obtain the electrostatic potential, dynamic pressure, tangentially induced electric field, and fluid velocity as functions of the lateral position in the slit in a self-consistent way, with the constraint of no net electric current arising from the cocurrent diffusion, electric migration, and diffusioosmotic convection of the electrolyte ions. The existence of the surface charge layers can lead to a diffusioosmotic flow quite different from that in a capillary with bare walls. The effect of the lateral distribution of the induced tangential electric field and the relaxation effect due to ionic convection in the slit on the diffusioosmotic flow are found to be very significant in practical situations.     

Diffusioosmosis of electrolyte solutions in a fine capillary tube

A theoretical study is presented for the steady diffusioosmotic flow of an electrolyte solution in a fine capillary tube generated by a constant concentration gradient imposed in the axial direction. The capillary wall may have either a constant surface potential or a constant surface charge density of an arbitrary quantity. The electric double layer adjacent to the charged wall may have an arbitrary thickness, and its electrostatic potential distribution is determined by an analytical approximation to the solution of the Poisson-Boltzmann equation. Solving a modified Navier-Stokes equation with the constraint of no net electric current arising from the cocurrent diffusion, electric migration, and diffusioosmotic convection of the electrolyte ions, the macroscopic electric field and the fluid velocity along the axial direction induced by the imposed electrolyte concentration gradient are obtained semianalytically as a function of the radial position in a self-consistent way. The direction of the diffusioosmotic flow relative to the concentration gradient is determined by the combination of the zeta potential (or surface charge density) of the wall, the properties of the electrolyte solution, and other relevant factors. For a prescribed concentration gradient of an electrolyte, the magnitude of fluid velocity at a position in general increases with an increase in its distance from the capillary wall, but there are exceptions. The effect of the radial distribution of the induced tangential electric field and the relaxation effect due to ionic convection in the double layer on the diffusioosmotic flow are found to be very significant.     

Diffusioosmosis and electroosmosis in a capillary slit with surface charge layers

This study analytically examines the steady diffusioosmotic and electroosmotic flows of an electrolyte solution in a fine capillary slit with each of its inside walls covered by a layer of adsorbed polyelectrolytes. In this solvent-permeable and ion-penetrable surface charge layer, idealized polyelectrolyte segments are assumed to distribute at a uniform density. The electric double layer and the surface charge layer may have arbitrary thicknesses relative to the gap width between the slit walls. The electrostatic potential distribution on a cross section of the slit is obtained by solving the linearized Poisson–Boltzmann equation, which applies to the case of low potentials or low fixed-charge densities. Explicit formulas for the fluid velocity profile due to the imposed electrolyte concentration gradient or electric field through the slit are derived as the solution of a modified Navier–Stokes/Brinkman equation. The results demonstrate that the structure of the surface charge layer can lead to an augmented or a diminished electrokinetic flow (even a reversal in direction of the flow) relative to that in a capillary with bare walls, depending on the characteristics of the capillary, of the surface charge layer, and of the electrolyte solution. For the diffusioosmotic flow with an induced electric field, competition between electroosmosis and chemiosmosis can result in more than one reversal in direction of the flow over a range of the Donnan potential of the adsorbed polyelectrolyte in the capillary.     

Electrokinetic diffusioosmotic flow of Ostwald-de Waele fluids near a charged flat plate in the thin double layer limit

Electrokinetic diffusioosmotic flow of Ostwald-de Waele, or power-law, fluids near a large charged flat plate is theoretically investigated for very thin double layers. Solutions to the flow velocity both up-close and far from the flat plate as well as the effective viscosity are presented for general values of the flow behavior index. Results show that given a wall zeta potential, ζ, diffusivity difference parameter, β, and constant imposed solute concentration gradient, both the near and far field diffusioosmotic flow velocities obtained for the respective dilatant and pseudoplastic liquids considerably deviate from those obtained for Newtonian liquids as found in previous literature. This likely suggests that the electrokinetic diffusioosmosis and its complementary effect of diffusiophoresis depend sensitively not only on the ζ-β parametric pair, but also on the possible non-Newtonian characteristics of the electrolytic liquid phase of the system. The theory presented herein can also be readily modified to model or describe electrodiffusioosmosis in power-law fluids, which is likely found in flow situations where the fluid non-Newtonian response, imposed solute concentration gradient, and an additional externally applied electric current density (or electric field) are of equal importance.     

Boundary effects on electrophoresis of a colloidal cylinder with a nonuniform zeta potential distribution

The electrophoretic motion of a long dielectric circular cylinder with a general angular distribution of its surface potential under a transversely imposed electric field in the vicinity of a large plane wall parallel to its axis is analyzed. The thickness of the electric double layers adjacent to the solid surfaces is assumed to be much smaller than the particle radius and the gap width between the surfaces, but the applied electric field can be either perpendicular or parallel to the plane wall. The presence of the confining wall causes three basic effects on the particle velocity: (1) the local electric field on the particle surface is enhanced or reduced by the wall; (2) the wall increases viscous retardation of the moving particle; (3) an electroosmotic flow of the suspending fluid may exist due to the interaction between the charged wall and the tangentially imposed electric field. Through the use of cylindrical bipolar coordinates, the Laplace and Stokes equations are solved analytically for the two-dimensional electric potential and velocity fields, respectively, in the fluid phase, and explicit formulas for the quasisteady electrophoretic and angular velocities of the cylindrical particle are obtained. To apply these formulas, one has only to calculate the multipole moments of the zeta potential distribution at the particle surface. It is found that the existence of a plane wall near a nonuniformly charged particle can cause its translation or rotation which does not occur in an unbounded fluid with the same applied electric field.     

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.     

Electrophoresis of a colloidal sphere in a spherical cavity with arbitrary zeta potential distributions

An analytical study is presented for the quasi-steady electrophoretic motion of a dielectric sphere situated at the center of a spherical cavity when the surface potentials are arbitrarily nonuniform. The applied electric field is constant, and the electric double layers adjacent to the solid surfaces are assumed to be much thinner than the particle radius and the gap width between the surfaces. The presence of the cavity wall causes three basic effects on the particle velocity: (1) the local electric field on the particle surface is enhanced or reduced by the wall; (2) the wall increases the viscous retardation of the moving particle; and (3) a circulating electroosmotic flow of the suspending fluid exists because of the interaction between the electric field and the charged wall. The Laplace and Stokes equations are solved analytically for the electric potential and velocity fields, respectively, in the fluid phase, and explicit formulas for the electrophoretic and angular velocities of the particle are obtained. To apply these formulas, one has to calculate only the monopole, dipole, and quadrupole moments of the zeta-potential distributions at the particle and cavity surfaces. It is found that the contribution from the electroosmotic flow developing from the interaction of the imposed electric field with the thin double layer adjacent to the cavity wall and the contribution from the wall-corrected electrophoretic driving force to the particle velocities can be superimposed as a result of the linearity of the problem.     

Diffusiophoresis in a suspension of spherical particles with arbitrary double-layer thickness

The diffusiophoresis in a homogeneous suspension of identical dielectric spheres with an arbitrary thickness of the electric double layers in a solution of a symmetrically charged electrolyte with a constant imposed concentration gradient is analytically studied. The effects of particle interactions (or particle volume fraction) are taken into account by employing a unit cell model, and the overlap of the double layers of adjacent particles is allowed. The electrokinetic equations that govern the ionic concentration distributions, the electrostatic potential profile, and the fluid flow field in the electrolyte solution surrounding the charged sphere in a unit cell are linearized assuming that the system is only slightly distorted from equilibrium. Using a perturbation method, these linearized equations are solved with the surface charge density (or zeta potential) of the particle as the small perturbation parameter. Analytical expressions for the diffusiophoretic velocity of the dielectric sphere in closed form correct to the second order of its surface charge density or zeta potential are obtained from a balance between its electrostatic and hydrodynamic forces. Comparisons of the results of the cell model with different conditions at the outer boundary of the cell are made.     

Particle Interactions in Diffusiophoresis and Electrophoresis of Colloidal Spheres with Thin but Polarized Double Layers

The diffusiophoretic and electrophoretic motions of two colloidal spheres in the solution of a symmetrically charged electrolyte are analyzed using a method of reflections. The particles are oriented arbitrarily with respect to the electrolyte gradient or the electric field, and they are allowed to differ in radius and in zeta potential. The thickness of the electric double layers surrounding the particles is assumed to be small relative to the radius of each particle and to the gap width between the particles, but the effect of polarization of the mobile ions in the diffuse layer is taken into account. A slip velocity of fluid and normal fluxes of solute ions at the outer edge of the thin double layer are used as the boundary conditions for the fluid phase outside the double layers. The method of reflections is based on an analysis of the electrochemical potential and fluid velocity disturbances produced by a single dielectric sphere placed in an arbitrarily varying electrolyte gradient or electric field. The solution for two-sphere interactions is obtained in expansion form correct to O(r(12)(-7)), where r(12) is the distance between the particle centers. Our analytical results are found to be in good agreement with the available numerical solutions obtained using a boundary collocation method. On the basis of a model of statistical mechanics, the results of two-sphere interactions are used to analytically determine the first-order effect of the volume fraction of particles of each type on the mean diffusiophoretic and eletrophoretic velocities in a bounded suspension. For a suspension of identical spheres, the mean diffusiophoretic velocity can be decreased or increased as the volume fraction of the particles is increased, while the mean electrophoretic velocity is reduced with the increase in the particle concentration. Generally speaking, the particle interaction effects can be quite significant in typical situations. Copyright 2000 Academic Press.     

Analysis of self-electrophoretic motion of a spherical particle in a nanotube: effect of nonuniform surface charge density

Autonomous motions of a spherical nanoparticle in a nanotube filled with an electrolyte solution were investigated using a continuum theory, which consisted of the Nernst-Planck equations for the ionic concentrations, the Poisson equation for the electric potential in the solution, and the Stokes equation for the hydrodynamic field. Contrary to the usual electrophoresis, in which an external electric field is imposed to direct the motion of charged particles, the autonomous motion originates from the self-generated electric field due to the ionic concentration polarization of the liquid medium surrounding an asymmetrically charged particle. In addition to the particle motion, the interaction between the electric field generated and the free charges of the polarized solution induces electroosmotic flows. These autonomous motions of the fluid as well as the particle were examined with focus on the effects of the surface-charge distribution of the particle, the size of the nanotube, and the thickness of the electric double layer, which affected the direction and the speed of the particle significantly.     

Diffusiophoresis of an Elongated Cylindrical Nanoparticle along the Axis of a Nanopore

The translation of a charged, elongated cylindrical nanoparticle along the axis of a nanopore driven by an imposed axial salt concentration gradient is investigated using a continuum theory, which consists of the ionic mass conservation equations for the ionic concentrations, the Poisson equation for the electric potential in the solution, and the modified Stokes equations for the hydrodynamic field. The diffusiophoretic motion is driven by the induced electrophoresis and chemiphoresis. The former is driven by the generated overall electric field arising from the difference in the ionic diffusivities and the double layer polarization, while the latter is generated by the induced osmotic pressure gradient around the charged particle. The induced diffusiophoretic motion is investigated as functions of the imposed salt concentration gradient, the ratio of the particle’s radius to the double layer thickness, the cylinder’s aspect ratio (length/radius), the ratio of the nanopore size to the particle size, the surface charge densities of the nanoparticle and the nanopore, and the type of the salt used. The induced diffusiophoretic motion of a nanorod in an uncharged nanopore is mainly governed by the induced electrophoresis, driven by the induced electric field arising from the double layer polarization. The induced particle motion is driven by the induced electroosmotic flow, if the charges of the nanorod and nanopore wall have the same sign.     

Diffusiophoresis of charged particles and diffusioosmosis of electrolyte solutions

A review is presented on the theoretical basics and recent developments about the diffusiophoresis of charged particles and diffusioosmosis of electrolyte solutions driven by imposed electrolyte concentration gradients with particular emphasis on the principal analytical formulas and their physical interpretations. For diffusiophoresis, migrations of particles with thin polarized electric double layers but arbitrary zeta potentials and with arbitrary double layers but relatively low surface potentials are both discussed in detail, covering not only diffusiophoresis of single particles but also their motions near solid boundaries or other particles. For diffusioosmosis, fluid flows along single plane walls, in micro/nano-channels, and in porous media are considered, in which the solid walls may have arbitrary zeta potentials or surface charge densities, and both the effect of the lateral distribution of the diffuse ions and the relaxation effect in the double layers on the tangential electric field induced by the prescribed electrolyte concentration gradient are included.     

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.     

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

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

Double-layer interaction between parallel solid cylinders partially immersed in an oil/water interface

We consider two identical, parallel, infinitely long solid cylinders at a given separation, lying flat on a plane oil/water interface and both immersed to the same extent in the oil and water phases. The part of the surface of each cylinder in contact with the aqueous phase is charged, forming an electric double layer in a symmetrical aqueous binary electrolyte. The electrical potential in the overlapping electric double layers in the aqueous phase satisfies the Poisson-Boltzmann equation. The potentials within the uncharged interiors of the solid cylinders and in the oil phase satisfy Laplace's equation. The equations for the three potentials are solved simultaneously using the finite element method with Galerkin weighted residuals. The double-layer interaction per unit length of the cylinders is then calculated. Of the numerical results obtained, three deserve special mention. First, a short-range double-layer repulsion, decaying exponentially with separation between the two cylinders, acts through the aqueous electrolyte medium, whereas in the case of an uncharged oil/water interface a weaker, but much longer-ranging, repulsive interaction acts through the oil medium. Second, reasonable estimates of the short-range interaction between cylinders in a planar interface can be obtained from the Derjaguin approximation for thin double layers. Third, in addition to the repulsive force between the cylinders parallel to the oil/water interface, a force normal to the interface acts on the cylinders in the direction of the aqueous electrolyte phase.     

Diffusiophoresis in a suspension of charge-regulating colloidal spheres

An analytical study of diffusiophoresis in a homogeneous suspension of identical spherical charge-regulating particles with an arbitrary thickness of the electric double layers in a solution of a symmetrically charged electrolyte with a uniform prescribed concentration gradient is presented. The charge regulation due to association/dissociation reactions of ionogenic functional groups on the particle surface is approximated by a linearized regulation model, which specifies a linear relationship between the surface charge density and the surface potential. The effects of particle-particle electrohydrodynamic interactions are taken into account by employing a unit cell model, and the overlap of the double layers of adjacent particles is allowed. The electrokinetic equations that govern the electric potential profile, the ionic concentration distributions, and the fluid flow field in the electrolyte solution surrounding the particle in a unit cell are linearized assuming that the system is only slightly distorted from equilibrium. Using a regular perturbation method, these linearized equations are solved with the equilibrium surface charge density (or zeta potential) of the particle as the small perturbation parameter. Closed-form formulas for the diffusiophoretic velocity of the charge-regulating sphere correct to the second order of its surface charge density or zeta potential are derived. Our results indicate that the charge regulation effect on the diffusiophoretic mobility is quite sensitive to the boundary condition for the electric potential specified at the outer surface of the unit cell. For the limiting cases of a very dilute suspension and a very thin or very thick electric double layer, the particle velocity is independent of the charge regulation parameter.     

Influence of viscoelectric effect on diffusioosmotic transport in nanochannel

We have investigated the role of viscoelectric effect on diffusioosmotic flow (DOF) through a nanochannel connected with two reservoirs. The transport equations governing the flow dynamics are solved numerically using the finite element technique. We have extensively analyzed the variation of induced field due to electric double layer (EDL) phenomenon, relative viscosity as modulated by the viscoelectric effect as well as reservoir's concentration difference, and their eventual impact on the underlying flow characteristics. It is revealed that the induced electric field in the EDL enhances fluid viscosity substantially near the charged wall at a higher concentration. We have shown that neglecting viscoelectric effect in the paradigm of diffusioosmotic transport overestimates the net throughput, particularly at a higher concentration difference. Furthermore, we show that pertaining to chemiosmosis dominated regime, the average flow velocity modifies with the increase in concentration difference up to a critical value. In comparison, the rise in the strength of resistive electroosmotic actuation by the accumulation of anions in the upstream reservoir reduces the average flow velocity at a higher concentration difference. We have reported a reduction in critical concentration with the increase in viscoelectric effect. The inferences of this analysis are deemed pertinent to reveal the bearing of viscoelectric effect as a flow control mechanism pertaining to DOF at nanoscale.