Mass transfer of a neutral solute in polyelectrolyte grafted soft nanochannel with porous wall

A soft nanochannel involves a soft interface that contains a polyelectrolyte layer (PEL) sandwiched between a rigid surface and a bulk electrolyte solution. Mass transfer of a neutral solute in a combined electroosmotic and pressure driven flow through a polyelectrolyte grafted charged nanochannel with porous wall is presented in this work. Assuming the PEL as fixed charged layer and PEL-electrolyte interface as a semi-penetrable membrane, analytical solutions were obtained for potential distributions (for small wall potential). Velocity profiles were also derived in the same domains, for both inside and outside the PEL. Convective-diffusive species balance equation was semi-analytically solved inside the PEL. Expression of length averaged Sherwood number was also obtained and effects of different parameters, namely, drag parameter (α), Debye parameter , and PEL thickness were studied in detail. The variation of permeate concentration and permeation flux across the porous wall was obtained.     

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.     

Temperature gradient focusing of bio-analyte in a microfluidic channel dealing with non-Newtonian electrolyte considering temperature-dependent zeta potential

Temperature gradient focusing (TGF) relies on establishing a precise balance between the electrophoretic motility of a target analyte and the advective flow of the background electrolyte (BGE) to locally concentrate the analyte in a microfluidic configuration. This paper presents a finite-element-based numerical analysis where the coupled electric field and the transport equations are solved to describe the effects of the shear-dependent apparent viscosity of a non-Newtonian BGE on the localized concentration buildup of a charged bio-sample inside a microchannel by TGF via Joule heating. Effects of the temperature-dependent nature of the wall zeta potential and the flow behavior index (n) of BGE on the flow, thermal, and species concentration profiles inside the microchannel have been investigated. Study using a fluorescein-Na analyte sample shows that the maximum normalized analyte concentration (C_max/C₀) reduces as the zeta potential increases linearly with temperature. The maximum concentration enhancement is achieved when the BGE displays the Newtonian rheology. For example, C_max/C₀ increases 134- to 280-fold when n is increased from 0.8 to 1 (pseudoplastic regime) and again reduces to 190-fold when n increases further from 1 to 1.2 (dilatant regime).     

Electrophoresis and electric conduction in a salt-free suspension of charged particles

The electrophoresis and electric conduction of a suspension of charged spherical particles in a salt-free solution are analyzed by using a unit cell model. The linearized Poisson-Boltzmann equation (valid for the cases of relatively low surface charge density or high volume fraction of the particles) and Laplace equation are solved for the equilibrium electric potential profile and its perturbation caused by the imposed electric field, respectively, in the fluid containing the counterions only around the particle, and the ionic continuity equation and modified Stokes equations are solved for the electrochemical potential energy and fluid flow fields, respectively. Explicit analytical formulas for the electrophoretic mobility of the particles and effective electric conductivity of the suspension are obtained, and the particle interaction effects on these transport properties are significant and interesting. The scaled zeta potential, electrophoretic mobility, and effective electric conductivity increase monotonically with an increase in the scaled surface charge density of the particles and in general decrease with an increase in the particle volume fraction, keeping each other parameter unchanged. Under the Debye-Hückel approximation, the dependence of the electrophoretic mobility normalized with the surface charge density on the ratio of the particle radius to the Debye screening length and particle volume fraction in a salt-free suspension is same as that in a salt-containing suspension, but the variation of the effective electric conductivity with the particle volume fraction in a salt-free suspension is found to be quite different from that in a suspension containing added electrolyte.     

Line of charges in electrolyte solution near a half-space II. Electric field of a single charge

The electric potential of a single charge in electrolyte solution near a dielectric or a semiconducting half-space is determined in closed form when the electrostatics is described by the linear Debye-Hückel (D-H) equation. The electric potential strongly depends on the Debye length of the solution, the substrate-to-solution dielectric constant ratio, and the Debye length of the semiconductor. The technique of Hankel transforms is shown to be a useful tool in solving such axially symmetric boundary value problems for the D-H equation.     

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.     

Electrohydrodynamic transport of non‐symmetric electrolyte through porous wall of a microtube

Transport of salt through the wall of porous microtube is relevant in various physiological microcirculation systems. Transport phenomena based modeling of such system is undertaken in the present study considering a combined driving force consisting of pressure gradient and external electric field. Transport of salt is modeled in two domains, in the flow conduit and in the pores of porous wall of the microtube. The solute transport in the microtube is presented by convective‐diffusive mass balance and it is solved using integral method under the framework of boundary layer analysis. The wall of the microtube is considered to be consisting of series of straight parallel cylindrical pores with charged inner surface. The solute transport through the pores is considered to be composed of diffusive, convective and electric potential gradient governed by Nernst‐Planck equation. Transport in the microtube and pores is coupled through the osmotic pressure model for the solvent and Donnan equilibrium distribution for the solute. The simulated results agree remarkably well with the experimental data conducted by in‐house experimental set up. The charge density of the porous wall is estimated through the minimization of errors involved between the experimental and simulated data for different operating conditions.     

The efficiency of electrokinetic pumping at a condition of maximum work

Numerical methods are employed to examine the work, electric power input, and efficiency of electrokinetic pumps at a condition corresponding to maximum pump work. These analyses employ the full Poisson-Boltzmann equations and account for both convective and conductive electric currents, including surface conductance. We find that efficiencies at this condition of maximum work depend on three dimensionless parameters, the normalized zeta potential, normalized Debye layer thickness, and a fluid property termed the Levine number indicating the nominal ratio of convective to conductive electric currents. Efficiencies at maximum work exhibit a maximum for an optimum Debye layer thickness when the zeta potential and Levine number are fixed. This maximum efficiency increases with the square of the zeta potential when the zeta potential is small, but reaches a plateau as the zeta potential becomes large. The maximum efficiency in this latter regime is thus independent of the zeta potential and depends only on the Levine number. Simple analytical expressions describing this maximum efficiency in terms of the Levine number are provided. Geometries of a circular tube and planar channel are examined.     

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.     

Combined electroosmotic and pressure-driven transport of neutral solutes across a rough,porous-walled microtube

A number of microfluidic systems of interest essentially consist of micro-scaled channels/tubes, whose walls are inherently rough. The novelty of the current study lies in exploring the impact of the wall roughness on mass transfer in the case of flow through a microtube with porous wall. The current investigation is possibly the first attempt at exploring the effect of mass transfer for a porous-walled, rough microtube, as earlier studies were limited to the analysis of hydrodynamic and thermal effects only in an impervious microtube. In particular, the effects of the corrugation amplitude and the wavenumber on the mass transport have been assessed in detail in this work, via a combination of perturbation approximations and numerical analysis. Several interesting revelations are elicited regarding the effects of these pertinent parameters on the mass transfer coefficient, permeation flux, wall surface concentration, and delivery flux of the neutral solute. It has been unveiled that it is possible to enhance the solute mass flux by 10% via appropriate tuning of corrugation amplitude. The findings of the study can help in better understanding of mass transport for a porous-walled, rough microtube, which has critical relevance in several important applications such as micromixers, targeted drug delivery, and so on.     

The rheology of pseudoplastic fluids in porous media using network modeling

This paper considers the rheology of pseudoplastic (shear thinning) fluids in porous media. The central problem studied is the relationship between the viscometric behavior of the polymer solution and its observed behavior in the porous matrix. In the past, a number of macroscopic approaches have been applied, usually based on capillary bundle models of the porous medium. These simplified models have been used along with constitutive equations describing the fluid behavior (usually of power law type) to establish semiempirical macroscopic equations describing the flow of non-Newtonian fluids in porous media. This procedure has been reasonably successful in correlating experimental results on the flow of polymer solutions through both consolidated and unconsolidated porous materials. However, it does not allow an interpretation of polymer flow in porous media in terms of the flows on a microscopic scale; nor does it allow us to predict changes in macroscopic behavior resulting from variations at a microscopic level in the characteristics of the porous medium such as pore size distribution. In this work, we use a network approach to the modeling of non-Newtonian rheology, in order to understand some of the more detailed features of polymjer flow in porous media. This approach provides a mathematical bridge between the behavior of the non-Newtonian fluid in a single capillary and the macroscopic behavior as deduced from the pressure drop-flow rate relation across the whole network model. It demonstrates the importance of flow redistribution within the elements of the capillary network as the overall pressure gradient varies. As an example of a pseudoplastic fluid in a porous medium, we consider the flow of xanthan biopolymer. This polymer is important as a displacing fluid viscosifier in enhanced oil recovery applications and, for that reason, a considerable amount of experimental data has been published on the flow of xanthan solutions in various porous media.     

Numerical analysis of thermophoresis of charged colloidal particles in non-Newtonian concentrated electrolyte solutions

Thermophoresis of colloidal particles in aqueous media is more frequently applied in biomedical analysis with processed fluids as biofluids. In this work, a numerical analysis of the thermophoresis of charged colloidal particles in non-Newtonian concentrated electrolyte solutions is presented. In a particle-fixed reference frame, the flow field of non-Newtonian fluids has been governed by the Cauchy momentum equation and the continuity equation, with the dynamic viscosity following the power-law fluid model. The numerical simulations reveal that the shear-thinning effect of pseudoplastic fluids is advantageous to the thermophoresis, and the shear-thickening effect of dilatant fluids slows down the thermophoresis. Both the shear-thinning and shear-thickening effects of non-Newtonian fluids on a thermodiffusion coefficient are pronounced for the case when the thickness of electric double layer (EDL) surrounding a particle is moderate or thin. Finally, the reciprocal of the dynamic velocity at the particle surface is calculated to approximately estimate the thermophoretic behavior of a charged particle with moderate or thin EDL thickness.     

Transport properties of long straight nano-channels in electrolyte solutions: a systematic approach

The principle of local thermodynamic equilibrium is systematically employed for obtaining various transport properties of long straight nano-channels. The concept of virtual solution is used to describe situations of non-negligible overlap of diffuse parts of electric double layers (EDLs) in nano-channels. Generic expressions for a variety of transport properties of long straight nano-channels are obtained in terms of quasi-equilibrium distribution coefficients of ions and functionals of quasi-equilibrium distribution of electrostatic potential. Further, the Poisson-Boltzmann approach is used to specify these expressions for long straight slit-like nano-channels. In the approximation of non-overlapped diffuse parts of double electric layers in nano-channels, simple analytical expressions are obtained for the apparent electrophoretic mobilities of (trace) analytes of arbitrary charge as well as for the salt reflection coefficient (osmotic pressure), salt diffusion permeability and electro-viscosity (electrokinetic energy conversion). The approximate solutions are compared with the results of rigorous solution of non-linearized Poisson-Boltzmann equation, and the accuracy of approximation is shown to be typically excellent when the nano-channel half-height exceeds ca.3 Debye screening lengths. Due to non-negligible electrostatic adsorption of ions by nano-channels, the apparent electrophoretic mobilities of counter-ionic analytes in nano-channels are smaller than in micro-channels whereas those of co-ionic analytes are larger. This dependence on the charge is useful for the separation of analytes of close electrophoretic mobilities. The osmotic pressure is shown to be positive, negative or pass through maxima as a function of applied salt-concentration difference within a fairly narrow range of ratios of nano-channel height to the Debye screening length. The diffusion permeability of charged nano-channels to single salts is demonstrated (for the first time) to be typically larger than that of neutral nano-channels of the same dimensions due to electrical facilitation of salt diffusion.     

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.     

Modified McKay–Perring Equation and Its Two Particular Solutions

Zdanovskii’s rule is the simplest isopiestic molality relation of mixed electrolyte aqueous solutions and the McKay–Perring equation is a differentio-integral equation particularly suitable for calculating solute activity coefficients from isopiestic measurements. However, they have two unsolved problems, which have puzzled solution chemists for several decades: (1) Zdanovskii’s rule has been verified by precise isopiestic measurements. But, several scientists suggested that the rule contradicts the Debye–Hückel limiting law for extremely dilute unsymmetrical mixtures. (2) In the McKay–Perring equation, a solute activity coefficient is multiplied by a solute composition variable. Different scientists have suggested that the composition variable may be the total ionic strength, common ion concentration, total ionic concentration, or an additive function with arbitrary proportionality constants. But, the different choices of the composition variable may lead to different activity coefficient results. Here, I derive a modified McKay–Perring equation in which the composition variable has the exclusive physical meaning of total ionic concentration for mixed electrolyte solutions (or of total solute particle concentration for the mixed solutions containing nonelectrolyte solutes). I also demonstrate that Zdanovskii’s rule is consistent with the Debye–Hückel limiting law for extremely dilute unsymmetrical mixtures. I derive two particular solutions of the modified McKay–Perring equation: one for the systems obeying Zdanovskii’s rule and another for the systems obeying a limiting linear concentration rule. These theoretical results have been verified with literature experiments and model calculations.     

Transport of charged samples in fluidic channels with large zeta potentials

In this article, we present an analysis on the transport of charged samples through micro- and nanofluidic channels with large zeta potentials (|zeta| > (kBT)/e). Using the Method of Moments formulation, the diffusion-convection equation has been solved to evaluate the mean velocity and the dispersion of analyte bands in a parallel-plate device under electrokinetically- and pressure-driven flow conditions. The effect of electromigration induced by the lateral electric field within the Debye layer has been quantified in our work using a Peclet number (Pe t) based on the characteristic electrophoretic velocity of the solute molecules in the transverse direction. It has been shown that while the effects of transverse electromigration on analyte transport only depends on the product Pe t zeta* for |zeta*| = (ezeta)/kBT < 1, both these parameters independently affect the flow of charged species in large zeta potential systems. For a given value of Pe t zeta*, the mean velocity and the slug dispersivity can vary by as much as an order of magnitude in going from a small zeta potential system (|zeta*| < 1) to a channel with |zeta*| = 4.     

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.     

Electrokinetic motion of a charged colloidal sphere in a spherical cavity with magnetic fields

The magnetohydrodynamic (MHD) effects on the translation and rotation of a charged colloidal sphere situated at the center of a spherical cavity filled with an arbitrary electrolyte solution when a constant magnetic field is imposed are analyzed at the quasisteady state. The electric double layers adjacent to the solid surfaces may have an arbitrary thickness relative to the particle and cavity radii. Through the use of a perturbation method to the leading order, the Stokes equations modified with the electric∕Lorentz force term are dealt by using a generalized reciprocal theorem. Using the equilibrium double-layer potential distribution in the fluid phase from solving the linearized Poisson-Boltzmann equation, we obtain explicit formulas for the translational and angular velocities of the colloidal sphere produced by the MHD effects valid for all values of the particle-to-cavity size ratio. For the limiting case of an infinitely large cavity with an uncharged wall, our result reduces to the relevant solution for an unbounded spherical particle available in the literature. The boundary effect on the MHD motion of the spherical particle is a qualitatively and quantitatively sensible function of the parameters a∕b and κa, where a and b are the radii of the particle and cavity, respectively, and κ is the reciprocal of the Debye screening length. In general, the proximity of the cavity wall reduces the MHD migration but intensifies the MHD rotation of the particle.     

Electrophoresis of hydrophilic/hydrophobic rigid colloid with effects of relaxation and ion size

This article deals with a semi‐analytical study on the electrophoresis of charged spherical rigid colloid by considering the effects of relaxation and ion size. The particle surface is taken to be either hydrophilic or hydrophobic in nature. In order to consider the ion size effect we have invoked the Carnahan and Starling model (J. Chem. Phys. 1969, 51, 635‐636). The mathematical model is based on Stokes equation for fluid flow, modified Boltzmann equation for spatial distribution of ionic species and Poisson equation for electric potential. We adopt a linear perturbation technique under a weak electric field assumption. An iterative numerical technique in employed to solve the coupled set of perturbed equations. We have validated the numerically obtained electrophoretic mobility with the corresponding analytical solution derived under low potential limit. Going beyond the widely employed Debye‐Hückel linearization, we have presented the results for a wide range of surface charge density, electrolyte concentration, and slip length to Debye length ratio. We have also identified several interesting features including occurrence of local maxima and minima in the mobility for critical choice of pertinent parameters.     

Changes in zeta potential caused by a dc electric current for thin double layers

An asymptotic solution was obtained to describe one-dimensional, steady-state transport of a symmetric binary electrolyte normal to two large parallel electrodes, in the limit in which the Debye length is infinitesimal compared to the distance separating the two electrodes. Despite the nonzero ion flux, Boltzmann's equation continues to describe the relationship between either ion concentration and the electrostatic potential inside the diffuse part of the double layer, while local electroneutrality applies outside, even for current densities approaching the limiting value. In the absence of ion adsorption or dissociation reactions at the electrodes, the magnitude of any charge or zeta potential arising on the electrodes at zero current is determined by the equilibrium constant for the redox reactions which would exchange ionic charge carriers for electric charge carriers at the electrode surface. Nonzero current causes the ionic strength of the bulk to vary with position. This perturbs the Debye length of the diffuse cloud on either electrode: it is the local ionic strength just outside the cloud which determines the Debye length for that cloud. Nonzero current also changes the zeta potential. The dimensionless rate of change dζ/dJ was as large as 30.