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 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.     

Electrokinetic flow in a capillary with a charge-regulating surface polymer layer

An analytical study of the steady electrokinetic flow in a long uniform capillary tube or slit is presented. The inside wall of the capillary is covered by a layer of adsorbed or covalently bound charge-regulating polymer in equilibrium with the ambient electrolyte solution. In this solvent-permeable and ion-penetrable surface polyelectrolyte layer, ionogenic functional groups and frictional segments are assumed to distribute at uniform densities. The electrical potential and space charge density distributions in the cross section of the capillary are obtained by solving the linearized Poisson-Boltzmann equation. The fluid velocity profile due to the application of an electric field and a pressure gradient through the capillary is obtained from the analytical solution of a modified Navier-Stokes/Brinkman equation. Explicit formulas for the electroosmotic velocity, the average fluid velocity and electric current density on the cross section, and the streaming potential in the capillary are also derived. The results demonstrate that the direction of the electroosmotic flow and the magnitudes of the fluid velocity and electric current density are dominated by the fixed charge density inside the surface polymer layer, which is determined by the regulation characteristics such as the dissociation equilibrium constants of the ionogenic functional groups in the surface layer and the concentration of the potential-determining ions in the bulk solution.     

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.     

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.     

Equilibrium aspects of polycation adsorption on silica surface: how the adsorbed layer responds to changes in bulk solution

Adsorption of cationic high molecular weight polyacrylamides (CPAM) (M(w) is about 800 kDa) with different fractions of cationic units tau = 0.09 and tau = 0.018 onto silica surface was studied over a wide range of pH (4-9) and KCl concentration (c(s) = 10(-3)-10(-1) M) by in-situ null ellipsometry. We discuss how the adsorbed layer depends on the bulk conditions as well as kinetically responds to changes in solution conditions. The adsorbed amount Gamma of CPAM increases with pH for all studied electrolyte concentrations until a plateau Gamma is reached at pH > 6. At low pH we observed an increase in adsorbed amount with electrolyte concentration. At high pH there is no remarkable influence of added salt on the values of the adsorbed amount. The thickness of adsorbed polymer layers, obtained by ellipsometry, increases with electrolyte concentration and decreases with pH. At low c(s) and high pH the polyelectrolyte adsorbs in a flat conformation. An overcompensation of the surface charge (charge reversal) by the adsorbed polyelectrolyte is observed at high c(s) and low pH. To reveal the reversibility of the polyelectrolyte adsorption with respect to the adsorbed amount and layer thickness, parameters such as polyelectrolyte concentration (c(p)), c(s), and pH were changed during the experiment. Generally, similar adsorbed layer properties were obtained independent of whether adsorption was obtained directly to initially bare surface or by changing pH, c(s), or the concentration of polyelectrolyte solution in the presence of a preadsorbed layer, provided that the coverage of the preadsorbed layer was low. Once a steady state of the measured parameters (Gamma, d) was reached, experimental conditions were restored to the original values and corresponding changes in Gamma and adsorbed layer thickness were recorded. For initially low surface coverage it was impossible to restore the layer properties, and in this case we always ended up with higher coverage than the initial values. For initial high surface coverage it was usually possible to restore the initial layer properties. Thus, we concluded that polyelectrolyte appears only partially reversible to changes in the solution conditions due the slow rearrangement process within the adsorbed layer.     

Electrokinetic flow and electric conduction of salt-free solutions in a capillary

The electrokinetic flow and accompanied electric conduction of a salt-free solution in the axial direction of a charged circular capillary are analyzed. No assumptions are made about the surface charge density (or surface potential) and electrokinetic radius of the capillary, which are interrelated. The Poisson–Boltzmann equation and modified Navier–Stokes equation are solved for the electrostatic potential distribution and fluid velocity profile, respectively. Closed-form formulas for the electroosmotic mobility and electric conductivity in the capillary are derived in terms of the surface charge density. The relative surface potential, electroosmotic mobility, and electric conductivity are monotonic increasing functions of the surface charge density and electrokinetic radius. However, the rises of the relative surface potential and electroosmotic mobility with an increase in the surface charge density are suppressed substantially when it is high due to the effect of counterion condensation. The analytical prediction that the electroosmotic mobility grows with increases in the surface charge density and electrokinetic radius agrees with the experimental results for salt-free solutions in circular microchannels in the literature.     

Cationic electrolyte adsorption layers on hydrophilic and hydrophobic surfaces

The capillary electrokinetics method (measurements of streaming potential and current in original and hydrophobized fused quartz capillaries with radii of 5–7 μm) is employed to study the formation of adsorption layers upon contact with solutions containing a cationic polyelectrolyte, poly(diallyldimethylammonium chloride). It is shown that polyelectrolyte adsorption causes the charge reversal of both hydrophilic and hydrophobic surfaces, with a smaller amount of the substance being adsorbed on the hydrophobic than on the hydrophilic surface. The adsorption on both surfaces increases with the polymer solution concentration. The cationic polyelectrolyte adsorption on the pure quartz surface occurs mainly due to the electrostatic attraction, while, in the case of the hydrophobic surface, the contribution of hydrophobic interactions increases. The study of the layer deformability shows that, on the hydrophilic surfaces, the layer ages and its structure depends on the polymer solution concentration. On the modified surface, the deformation of even freshly formed layers is slight, which suggests that a denser layer is formed on the hydrophobic surface. In contrast to the hydrophilic surface, the polyelectrolyte is partly desorbed from the hydrophobic surface.     

Electroosmotic Flow through an Annulus

The electro-osmosis through an annulus is investigated. The electric potential and flow velocity profile are obtained by solving the linearized Poisson-Boltzmann equation and the Stokes equation. Both the thin and thick double layer limits are analyzed. Under the condition of thin double layer, the electro-osmotic mobility can be described by the Helmholtz-Smoluchowski equation with a geometry-dependent correction factor. There exist net flows even for zero area-averaged surface charge density due to the curvature differences between the inner and outer walls. The flow direction is determined by the sign of the charge on the inner cylinder. We also found that under certain circumstances the flow direction in an annulus is opposite to that in a capillary with the same sign of the net charge. Copyright 2000 Academic Press.     

Poisson-Boltzmann theory of the charge-induced adsorption of semi-flexible polyelectrolytes

A model is suggested for the structure of an adsorbed layer of a highly charged semi-flexible polyelectrolyte on a weakly charged surface of opposite charge sign. The adsorbed phase is thin, owing to the effective reversal of the charge sign of the surface upon adsorption, and ordered, owing to the high surface density of polyelectrolyte strands caused by the generally strong binding between polyelectrolyte and surface. The Poisson-Boltzmann equation for the electrostatic interaction between the array of adsorbed polyelectrolytes and the charged surface is solved for a cylindrical geometry, both numerically, using a finite element method, and analytically within the weak curvature limit under the assumption of excess monovalent salt. For small separations, repulsive surface polarization and counterion osmotic pressure effects dominate over the electrostatic attraction and the resulting electrostatic interaction curve shows a minimum at nonzero separations on the Angstrom scale. The equilibrium density of the adsorbed phase is obtained by minimizing the total free energy under the condition of equality of chemical potential and osmotic pressure of the polyelectrolyte in solution and in the adsorbed phase. For a wide range of ionic conditions and charge densities of the charged surface, the interstrand separation as predicted by the Poisson-Boltzmann model and the analytical theory closely agree. For low to moderate charge densities of the adsorbing surface, the interstrand spacing decreases as a function of the charge density of the charged surface. Above about 0.1 M excess monovalent salt, it is only weakly dependent on the ionic strength. At high charge densities of the adsorbing surface, the interstrand spacing increases with increasing ionic strength, in line with the experiments by Fang and Yang [J. Phys. Chem. B 101, 441 (1997)].     

Electrophoresis of soft particles with charged rigid core coated with pH-regulated polyelectrolyte layer

We present a theoretical study on the electrophoresis of a soft particle with a dielectric charged rigid core grafted with a charge-regulated polyelectrolyte layer. The polyelectrolyte layer possesses either an acidic or a basic functional group and the charge dissociation depends on the local pH and ionic concentration of the electrolyte. The dielectric rigid core is considered to possess a uniform volumetric charge density. The electric potential distribution is determined by computing the Poisson-Boltzmann equation outside the core coupled with a Poisson equation inside the impermeable core along with suitable matching conditions at the core-shell interface. The computed electric field is used to determine the mobility of the particle through an existing analytic expression based on the Debye-Huckel approximation. Our results are found to be in good agreement with the existing solutions for the limiting cases. The influence of the core charge density, ionic concentration, and pH of the electrolyte on the particle mobility is studied for different choice of hydrodynamic penetration length of the polyelectrolyte and dissociation constant of the functional group. The critical value of the pH required to achieve zero mobility is estimated. We find that in a monovalent electrolyte solution, the soft particle with a net negative (positive) charge can have positive (negative) mobility.     

Diffusiophoresis and electrophoresis of a charged sphere parallel to one or two plane walls

The diffusiophoretic and electrophoretic motions of a dielectric spherical particle in an electrolyte solution located between two infinite parallel plane walls are studied theoretically. The imposed electrolyte concentration gradient or electric field is constant and parallel to the two plates, which may be either impermeable to the ions/charges or prescribed with the far-field concentration/potential distribution. The electrical double layer at the particle surface is assumed to be thin relative to the particle radius and to the particle-wall gap widths, but the polarization effect of the mobile ions in the diffuse layer is incorporated. The presence of the neighboring walls causes two basic effects on the particle velocity: first, the local electrolyte concentration gradient or electric field on the particle surface is enhanced or reduced by the walls, thereby speeding up or slowing down the particle; second, the walls increase the viscous retardation of the moving particle. To solve the conservative equations, the general solution is constructed from the fundamental solutions in both rectangular and spherical coordinates. The boundary conditions are enforced first at the plane walls by the Fourier transforms and then on the particle surface by a collocation technique. Numerical results for the diffusiophoretic and electrophoretic velocities of the particle relative to those of a particle under identical conditions in an unbounded solution are presented for various values of the relevant parameters including the relative separation distances between the particle and the two plates. For the special case of motions of a spherical particle parallel to a single plate and in the central plane of a slit, the collocation results agree well with the approximate analytical solutions obtained by using a method of reflections. The presence of the lateral walls can reduce or enhance the particle velocity, depending on the properties of the particle-solution system, the relative particle-wall separation distances, and the electrochemical boundary condition at the walls. In general, the boundary effects on diffusiophoresis and electrophoresis are quite significant and complicated, and they no longer vary monotonically with the separation distances for some situations.     

On the indirect polyelectrolyte titration of cellulosic fibers. Conditions for charge stoichiometry and comparison with ESCA

The effect of electrolyte (NaHCO3) concentration on the adsorption of poly-DADMAC (poly-diallyldimethylammonium chloride) onto cellulosic fibers with different charge profiles was investigated. Surface carboxymethylated fibers were obtained by grafting carboxymethyl cellulose (CMC) onto the fiber surface and bulk carboxymethylated fibers were obtained by reacting the fibers with monochloroacetic acid. It was shown that nonionic interactions do not exist between cellulose and poly-DADMAC, rather electrostatic interactions govern the adsorption. Charge stoichiometry prevails under electrolyte-free conditions, whereas surface charge overcompensation occurs at higher electrolyte concentrations. It was shown that charge stoichiometry prevails if the thickness of the electric double layer kappa(-1) was larger than the mean distance between the charges on the fiber surface, as predicted by polyelectrolyte adsorption theories, taking lateral correlation effects into account. In a second set of experiments the ESCA technique served to independently calibrate the polyelectrolyte titrations for determining the surface charge of cellulosic fibers. Various molecular masses of poly-DADMAC were adsorbed to carboxymethylated fibers having different charge profiles. The adsorption of low M(w) poly-DADMAC (7.0 x 10(3)), analyzed by polyelectrolyte titration, was about 10 times higher than that of the high M(w) poly-DADMAC (9.2 x 10(5)). Despite the difference in accessibility of these two polyelectrolytes to the fiber cell wall, ESCA surface analysis showed, as expected, only slight differences between the two polyelectrolytes. This gives strong credibility to the idea that surface charge content of cellulosic fibers can be analyzed by means of adsorption of a high-molecular-mass cationic polymer, i.e., by polyelectrolyte titration.     

Flow of solutions in thin capillaries coated with an adsorbed polyelectrolyte lyer

The flow of KCl solutions through thin quartz capillaries coated with an adsorbed layer of a cationic polyelectrolyte (CPE), poly(dimethyldiallylammonium chloride) (molecular mass M = 500000), is studied. It is found that the adsorption layer is soft and its thickness depends on shear stress generated by the liquid flow through the capillary. The hydrodynamic thickness of the CPE adsorption layer is 80–90 nm at low flow rates of a solution, and it decreases to values comparable with the experimental error at high flow rates. The dried adsorption layer appears to be hydrophobic (the advancing contact angle is about 80°); in these capillaries, the flow rate of a KCl solution is increased that can be interpreted as a solution slip on the surface of CPE adsorption layer. The long-term contact of the dried CPE adsorption layer with KCl solution, probably, results in the swelling of the adsorption layer, which is accompanied by a decrease in the contact angle and ζ potential of the adsorption layer surface as calculated from the streaming potential of the same solution.     

Adsorption of low charge density polyelectrolytes to an oppositely charged porous substrate

The adsorption behavior of a low charge density cationic polyelectrolyte to cellulosic fibers has been studied. Cationic dextran served as a model polyelectrolyte, as it can be prepared over a range in molecular mass and charge density. The adsorption behavior of the cationic dextran was measured in electrolyte-free conditions using polyelectrolyte titration techniques. By fluorescent labeling the cationic dextran, the extent to which adsorption occurs inside the porous structure was further determined by fluorescent confocal laser scanning microscopy. Cationic dextran having a sufficiently low charge density adsorbed into the pores, although the extent the cationic dextran adsorbed was governed by the molecular mass. The adsorption behavior of the cationic dextran was also studied in various electrolyte concentrations. The adsorbed mass monotonically decreased with increasing electrolyte, as the electrostatic interaction with the substrate was more effectively screened. This behavior also suggests that the interactions between adsorbed polyelectrolyte chains, i.e. lateral correlation effects, are negligible for low charge density polyelectrolytes. Finally, the effect of having a preadsorbed layer of cationic dextran on the adsorption behavior was determined in electrolyte-free conditions using fluorescent double staining techniques. The preadsorbed cationic dextran had almost no effect on the adsorption of low molecular mass fractions. Low molecular mass fractions directly adsorbed into the pore structure, as opposed to adsorbing to a free surface and diffusing into the pores. It was also shown that cationic dextran can be selectively adsorbed to different locations, such that the surface of a porous substrate can be treated uniquely from the bulk.     

A modified Poisson-Boltzmann model including charge regulation for the adsorption of ionizable polyelectrolytes to charged interfaces, applied to lysozyme adsorption on silica

The equilibrium adsorption of polyelectrolytes with multiple types of ionizable groups is described using a modified Poisson-Boltzmann equation including charge regulation of both the polymer and the interface. A one-dimensional mean-field model is used in which the electrostatic potential is assumed constant in the lateral direction parallel to the surface. The electrostatic potential and ionization degrees of the different ionizable groups are calculated as function of the distance from the surface after which the electric and chemical contributions to the free energy are obtained. The various interactions between small ions, surface and polyelectrolyte are self-consistently considered in the model, such as the increase in charge of polyelectrolyte and surface upon adsorption as well as the displacement of small ions and the decrease of permittivity. These interactions may lead to complex dependencies of the adsorbed amount of polyelectrolyte on pH, ionic strength, and properties of the polymer (volume, permittivity, number, and type of ionizable groups) and of the surface (number of ionizable groups, pK, Stern capacity). For the adsorption of lysozyme on silica, the model qualitatively describes the gradual increase of adsorbed amount with pH up to a maximum value at pHc, which is below the iso-electric point, as well as the sharp decrease of adsorbed amount beyond pHc. With increasing ionic strength the adsorbed amount decreases (for pH > pHc), and pHc shifts to lower values.     

Discrete charge effects on the Donnan potential and surface potential of a soft particle

The Donnan potential and surface potential of soft particles (i.e., polyelectrolyte-coated hard particles) in an electrolyte solution play an essential role in their electric behaviors. These potentials are usually derived via a continuum model in which fixed charges inside the surface layer are distributed with a continuous charge density. In this paper, for a plate-like soft particle consisting of a cubic lattice of fixed point charges, on the basis of the linearized Poisson–Boltzmann equation, we derive expressions for the electric potential distribution in the regions inside and outside the surface layer. This expression is given in terms of a sum of the screened Coulomb potentials produced by the point charges within the surface layer. We show that the deviation of the results of the discrete charge model from those of the continuous charge model becomes significant as the ratio of the lattice spacing to the Debye length becomes large.     

Noncovalent coatings for the separation of synthetic polypeptides by nonaqueous capillary electrophoresis

Recently, we demonstrated the possibility to extend the range of capillary electrophoresis (CE) applications to the separation of non-water-soluble synthetic polymers. This work focuses on the control of the electro-osmotic flow (EOF) and on the limitation of the solute adsorption in nonaqueous electrolytes. For these purposes, different strategies were investigated. For the initial, a viscous additive (ethylene glycol or glycerol) was used in the electrolyte in order to decrease the EOF magnitude and, possibly, to compete with solute adsorption. A second strategy was to modify, before separation, the fused-silica capillary wall by the adsorption of poly(ethylene oxide) (PEO) via hydrogen bonding. The influence of the molecular mass of the adsorbed PEO on the EOF magnitude and direction was studied in electrolytes based on methanol/acetonitrile mixtures containing ammonium ions. For PEO molecular masses above 1000 g/mol, reversed (anodic) EOF were reported in accordance with previous results obtained with PEO covalently bonded capillaries. The influence of the nature and the concentration of the background electrolyte cation on the EOF magnitude and direction were also investigated. A third strategy consisted in modifying the capillary wall by the adsorption of a cationic polyelectrolyte layer. Advantageously, this polyelectrolyte layer suppressed the adsorption of the polymer solutes onto the capillary wall. The results obtained in this work confirm the high potential and the versatility of CE for the characterization of ionizable organic polymers in nonaqueous media.     

Diffusioosmotic flow in rectangular microchannels

The diffusioosmosis of an electrolyte solution inside a uniformly charged rectangular channel at steady locally developed conditions is the subject of this study. Utilizing a finite element based numerical procedure, we try to estimate the errors incurred by modeling the actual rectangular geometry of typical microchannels as a slit. We demonstrate that the flow pattern and direction are generally dependent upon the width‐to‐height ratio of the channel. Such a finding, besides showing the ineffectiveness of the slit geometry in representing a rectangular channel of small aspect ratio, informs us of another mechanism of controlling the diffusioosmotic flow. Inspections of the mean velocity reveal that, although it drastically grows by increasing the aspect ratio at smaller values of this parameter, no significant change is observed when the aspect ratio is 5 or higher. The same trend is observed when EDL is shrunk and is considered as a basis for the introduction of a slip‐like velocity, similar to the concept of the Helmholtz–Smoluchowski electroosmotic velocity, which will be of high practical importance when dealing with a micronsized channel. Because of its significance, an expression is presented for this slip velocity utilizing the curve fitting of the results, assuming a typical Peclet number.