Streaming Potential and Associated Electrokinetic Effects through a Channel Filled with Electrolyte Solution Surrounded by a Layer of Immiscible and Dielectric Liquid.

The present article deals with the streaming potential-mediated pressure-driven flow across a channel in which the electrolyte solution is surrounded by a layer of cell membrane. Such a membrane of a biological cell may be modeled as an immiscible and dielectric liquid, which may bear free lipid molecules or charged surfactants. The presence of such additional charged molecules may lead to formation of liquid-liquid interfacial charge. In addition, the dielectric gradient-mediated ion partitioning effect further plays an important role in two-phase electrokinetic motion. We aim to study the generation of streaming potential and electrokinetic conversion efficiency as well as associated electroviscous effect for the undertaken problem. The mathematical model is based on the Poisson-Boltzmann equation for electrostatic potential and the Stokes equation for fluid flow, and the problem is studied considering suitable interfacial conditions for the flow variables along the liquid-liquid interface. The explicit analytical results for velocity and streaming field, electrokinetic energy conversion efficiency, and the parameter indicating the electroviscous effect are derived under the Donnan limit and within the Debye-Hückel electrostatic framework. We further numerically calculated the aforementioned intrinsic electrokinetic parameter associated with the problem undertaken for a wide range of pertinent parameters. The results are illustrated to indicate the impact of pertinent parameters on the generation of the streaming potential and associated electrokinetic effects.

Electrohydrodynamics of diffuse porous colloids.

The present article deals with the electrohydrodynamic motion of diffuse porous particles governed by an applied DC electric field. The spatial distribution of monomers as well as the charge distribution across the particle are considered to follow sigmoidal distribution involving decay length. Such a parameter measures the degree of inhomogeneity of the monomer distribution across the particle. The diffuse porous particles resemble several colloidal entities which are often seen in the environment as well as in biological and pharmaceutical industries. Considering the impact of bulk pH and ion steric effects, we modelled the electrohydrodynamics of such porous particulates based on the modified Boltzmann distribution for the spatial distribution of electrolyte ions and the Poisson equation for electric potential as well as the conservation of mass and momentum principles. We adopt regular perturbation analysis with weak field assumption and the perturbed equations are solved numerically to calculate the electrophoretic mobility and neutralization fraction of the particle charge during its motion as well as fluid collection efficiency. We further deduced the closed form relation between the drag force experienced by the charged porous particle and the fluid collection efficiency. In addition to the numerical results, we further derived the closed form analytical results for all the intrinsic parameters indicated above derived within the Debye-Hückel electrostatic framework and homogeneous distribution of monomers within the particle for which the decay length vanishes. The deduced mathematical results as indicated above will be useful to analyze several electrostatic and hydrodynamic features of a wide class of porous particulate and environmental entities.

Effect of the Surface Charge-Dependent Boundary Slip on the Electrophoresis of a Hydrophobic Polarizable Rigid Colloid.

The electrophoresis of a hydrophobic charged rigid colloid is studied by considering the lateral movement of the adsorbed surface charge. The slip velocity condition at the hydrophobic surface is modified to take into account the impact of the frictional and electric forces created by the adsorbed laterally mobile surface charge. Though the dependency of the surface charge on the slip velocity in the context of electrophoresis has been addressed before, the effect of the laterally mobile adsorbed surface charge on the electrophoresis of hydrophobic colloids has not been studied. The dielectric colloid is considered to polarize and create an induced immobile surface charge when subjected to an imposed electric field. The impact of the mobile surface charge along with the immobile induced surface charge on electrophoresis of a hydrophobic colloid is elucidated by numerically solving the governing electrokinetic equations in their full form. We have also developed a simplified model under a weak applied field consideration, which can be further reduced to a closed-form analytic expression for the mobility under the Debye-Hückel approximation. This analytic model for mobility is in excellent agreement with the exact numerical solution for an entire range of the Debye length when the ζ-potential is in the order of the thermal potential. One of the notable features of this closed-form mobility expression is that it accounts for the mobile adsorbed surface charge on the hydrodynamic slip condition and the dielectric polarization of the particle. We find that the mobility of the surface charge decreases the electrophoretic mobility of the hydrophobic dielectric colloid. However, the mobile surface charge enhances the mobility of a conducting hydrophobic colloid.

Dynamic Electrophoresis of a Hydrophobic and Dielectric Fluid Droplet.

Dynamic electrophoresis is the foundation for electroacoustical measurements, in which the electroacoustical signals may be used to analyze the size and electrostatic charge of colloidal entities by means of the results for dynamic electrophoretic mobility. Thus, the electrophoresis under an alternating electric field is the key foundation for electroacoustic theory. In this article, we develop a tractable analytical theory for the dynamic electrophoresis of hydrophobic and dielectric fluid droplets possessing uniform surface charge density. The tiny fluid droplets possess charged mobile surfaces and have found widespread applications in our day-to-day life. For dielectric fluid droplets (e.g., oil-water emulsions), the tangential electric stress at the interface is nonzero, which significantly affects its electrohydrodynamics under an oscillatory electric field, which has, however, a negligible impact on the electrophoretic motion of conducting droplets (e.g., mercury droplets). Besides, the micro/nanoscale fluid droplets often show hydrophobicity when they are immersed in an aqueous medium, and the impact of the electric field on hydrophobic surfaces remains a research frontier in the chemical discipline. Whereas a number of approximate expressions for electrophoretic mobility have been derived for the conducting droplet, none of them are applicable to such generic hydrophobic fluid droplets with dielectric permittivity that is significantly lower than or comparable to that of an aqueous medium. In this work, within the Debye-Hückel electrostatic framework, we elaborate an original analytical expression of dynamic electrophoretic mobility for this generic dielectric fluid droplet with a hydrophobic surface considering that the droplet retains its spherical shape during its oscillatory motion. We further derived a set of simplified expressions for dynamic electrophoretic mobility deduced under several limiting cases. The results are further illustrated, indicating the impact of pertinent parameters.

Diffusiophoresis of Weakly Charged Fluid Droplets in a General Electrolyte Solution: An Analytical Theory.

Owing to the importance of analytical results for electrokinetics of colloidal entities, we performed a mathematical analysis to determine the closed form analytical results for the diffusiophoretic velocity of a hydrophobic and polarizable fluid droplet. A comprehensive mathematical model is developed for diffusiophoresis, considering the background aqueous medium as general electrolytes (e.g., binary valence-symmetric/asymmetric electrolytes and a mixed solution of binary electrolytes). We performed our analysis under a weak concentration gradient, and the analytical results for diffusiophoretic velocity are calculated within the Debye-Hückel electrostatic framework. The exact form of the diffusiophoretic velocity is further approximated with negligible error, and the approximate form is found to be free from any of the cumbersome exponential integrals and thus very convenient for practical use. The present theory also covers the diffusiophoresis of perfectly dielectric as well as perfectly conducting droplets as its limiting case. In addition, we have further derived a number of closed form expressions for diffusiophoretic velocity pertaining to several particular cases, and we observed that the derived limit correctly recovers the available existing analytical results for diffusiophoretic velocity. Thus, the present analytical theory for diffusiophoresis can be applied to a wide class of fluidic droplets, e.g., hydrophobic and dielectric oil/conducting mercury droplets, air bubbles, nanoemulsions, as well as any polarizable and hydrophobic fluidic droplet suspended in a solution of general electrolytes.

Electrophoresis of Dielectric and Hydrophobic Spherical Fluid Droplets Possessing Uniform Surface Charge Density.

The present article deals with the theoretical study on electrophoresis of hydrophobic and dielectric spherical fluid droplets possessing uniform surface charge density. Unlike the ideally polarizable liquid droplet bearing constant surface ζ-potential, the tangential component of the Maxwell stress is nonzero for dielectric fluid droplets with uniform surface charge density. We consider the continuity of the tangential component of total stress (sum of the hydrodynamic and Maxwell stresses) and jump in dielectric displacement along the droplet-to-electrolyte interface. The typical situation is considered here for which the interfacial tension of the fluid droplet is sufficiently high so that the droplet retains its spherical shape during its motion. The present theory can be applied to nanoemulsions, hydrophobic oil droplets, gas bubbles, droplets of immiscible liquid suspended in aqueous medium, etc. Based on weak field and low charge assumptions and neglecting the Marangoni effect, the resultant electrokinetic equations are solved using linear perturbation analysis to derive the closed form expression for electrophoretic mobility applicable for the entire range of Debye-Hückel parameter. We further deduced an alternate approximate expression for electrophoretic mobility without involving exponential integrals. Besides, we have derived analytical results for mobility pertaining to various limiting cases. The results are further illustrated to show the impact of pertinent parameters on the overall electrophoretic mobility.

Gel Electrophoresis of a Hydrophobic Liquid Droplet with an Equipotential Slip Surface.

A theoretical study has been carried out on the electrophoresis of charged dielectric liquid droplets with an equipotential and hydrodynamically slipping surface moving in a quenched polymeric charged hydrogel medium. The liquid inside the droplet is electrically neutral. The Brinkman-Debye-Bueche model is employed to study the gel electrophoresis of such a hydrophobic and equipotential liquid droplet considering the long-range hydrodynamic interaction between a migrating droplet and the gel skeleton. Within the weak field and Debye-Hückel electrostatic framework, we derive an original closed-form expression for electrophoretic mobility, which further recovers the existing mobility expressions derived under several limiting conditions. The derived expressions for electrophoretic mobility explicitly involve exponential integrals, which are not so convenient for practical applications. Thus, the exact forms of the electrophoretic mobility under various electrohydrodynamic conditions are further approximated to make them free from exponential integrals. The approximate forms are found to be in excellent agreement with the exact results with maximum relative errors of about 1.5%.

Electrophoresis of soft particles with hydrophobic inner core grafted with pH-regulated and highly charged polyelectrolyte layer.

Electrophoresis of core-shell composite soft particles possessing hydrophobic inner core grafted with highly charged polyelectrolyte layer (PEL) has been studied analytically. The PEL bears pH-dependent charge properties due to the presence of zwitterionic functional groups. The dielectric permittivity of the PEL and bulk aqueous medium were taken to be different, which resulted in the ion-partitioning effect. Objective of this study was to provide a simple expression for the mobility of such core-shell soft particles under Donnan limit where the thickness of the PEL well exceeds the electric double layer thickness. Going beyond the widely used Debye-Hückel linearization, the nonlinear Poisson-Boltzmann equation coupled with Stokes-Darcy-Brinkman equations was solved to determine the electrophoretic mobility. The derived expression further recovers all the existing results for the electrophoretic mobility under various simplified cases. The graphical presentation of the results illustrated the impact of pertinent parameters on the electrophoretic mobility of such a soft particle.

Electrophoresis of Liquid-Layer Coated Particles: Impact of Ion Partitioning and Ion Steric Effects.

The biomimetic core-shell nanoparticles coated with membranes of various biological cells have attracted significant research interest, because of their extensive applications in targeted drug delivery systems. The cell membrane consists of a lipid bilayer, which can be regarded as a two-dimensional oriented viscous liquid with low dielectric permittivity, compared to a bulk aqueous medium. Such a liquid layer comprised of cell membrane may bear additional mobile charges, because of the presence of free lipid molecules or charged surfactant molecules, which further results in nonzero charge along the surface of the peripheral layer. In this article, we present an analytical theory for electrophoresis of such cell membrane coated functionalized nanoparticles in the extent of electrolyte solution, considering the combined effects of finite ion size and of ion partitioning. Going beyond the Debye-Huckel approximations, we propose an analytical theory for Donnan potential and electrophoretic mobility. The derived expressions are applicable for moderate to highly charged undertaken core-shell particles when the thickness of the peripheral liquid layer greatly exceeds the electric double layer thickness. The impact of pertinent parameters on the electrophoretic response of such a particle is further discussed.

A simplified model for gel electrophoresis of a hydrophobic rigid colloid.

Electrophoresis of a charged dielectric hydrophobic colloid embedded in a charged hydrogel medium is addressed. A slip velocity condition at the particle surface is considered. The characteristic of the gel electrophoresis is different compared with the free-solution electrophoresis due to the presence of immobile charges of the gel medium, which induces a strong background electroosmotic flow and modifies the Debye layer of the colloid. The gel electrophoresis of the dielectric hydrophobic charged colloid is made based on first-order perturbation analysis. A closed form solution involving simple exponential integrals for the mobility is derived, which reduces to several existing mobility expressions under limiting conditions such as for the gel electrophoresis of hydrophilic particles and a hydrophobic colloid in free-solution electrophoresis. We find that the mobility reversal is achieved by varying the Debye length or gel permeability. For the present first-order perturbation analysis, unlike free-solution electrophoresis, the particle dielectric permittivity is found to influence the mobility. One of the intriguing features of the present study is the derivation of the simplified mobility expression, which can be easily computed for a given set of parameter values.

Effect of core hydrophobicity on the electrophoresis of pH-regulated soft particles.

We propose a theoretical study on the electrophoresis of core-shell composite soft particles considering the effect of hydrodynamic slip length of the hydrophobic inner core. The surface of the inner core as well as the soft polymeric shell bear zwitterionic functional groups and the charged conditions depend on the nearby micro-environment. Within a low potential and weak electric field framework, the mathematical equations of the generalized electrokinetic theory for soft surfaces are solved analytically subject to appropriate boundary conditions, and a general electrophoretic mobility expression in an integral form involving the pH-dependent electrostatic potential is derived. With the help of suitable numerical schemes, electrophoretic mobility can easily be obtained. The effect of hydrophobicity of the inner core on the electrophoretic mobility of pH-regulated soft particles is illustrated for a wide range of pertinent parameters.

Electrokinetics of Concentrated Suspension of Soft Particles with pH-Regulated Volumetric Charges.

This article presents a theoretical study on the electrokinetics of concentrated suspension of charge-regulated soft particles under a weak electric field and low potential assumptions. The inner core of the undertaken particle is "semisoft" in nature, which allows ion penetration while the fluid cannot flow within it, and the outer soft polymeric shell allows the flow of the ionized fluid. In addition, the inner core and the outer polyelectrolyte layer (PEL) bear pH-regulated basic and acidic functional groups, respectively. The Poisson-Boltzmann equation-based mathematical model was adopted here for electric potential. The fluid flow across the electrolyte medium and PEL is governed by the Stokes equation and the Darcy-Brinkman equation, respectively. The Kuwabara's unit cell model (J. Phys. Soc. Japan, 1959, 14, 522-527) was invoked to observe the effect of the interaction between the neighboring particles in a concentrated suspension. A first order perturbation technique was used to determine the mean electrophoretic mobility of the undertaken soft particles in a concentrated suspension. The effect of pH and concentration of bulk electrolyte, electrohydrodynamic properties of both the inner core and PEL, on the mean electrophoretic mobility has been studied extensively. In addition, the results have been presented for the neutralization factor that measures the fraction of fixed charges neutralized by the mobile counterions.

Settling of a charged hydrophobic rigid colloid in aqueous media under generalized gravitational field.

The hindrance created by the induced electric filed on the sedimentation of a charged colloid in an aqueous media is studied through numerical modeling. The colloid is considered to be hydrophobic, sedimenting under gravity or a centrifugal force (generalized gravity). The deformation of the charge cloud around the colloid induces an electric field, which generates electrical dipole force on the colloid. The sedimentation velocity is governed by the balance of an electric force, hydrodynamic drag, and gravitational force. Governing equations based on the first principle of electrokinetics is solved numerically through a control volume approach. The dependence of the sedimentation velocity on the electrical properties and slip length of the colloid is investigated. The sedimentation velocity of the charged colloid is slower than the corresponding uncharged particle and this deviation magnifies as the charge density as well as particle slip length is increased. An enhanced g-factor creates a size dependency of the charged colloids. The induced sedimentation field is obtained to analyze the electrokinetics. Surface hydrophobicity enhances the sedimentation velocity, which in turn manifests the induced sedimentation field. However, the sedimentation velocity of a charged hydrophobic colloid is lower than the corresponding uncharged hydrophobic particle and this deviation manifests as slip length is increased.

Electrophoresis of dielectric and immiscible-liquid-layer-encapsulated colloids in aqueous media.

In this paper we consider the electrophoresis of a functionalized nanoparticle in electrolyte solution. The undertaken particle is comprised of a rigid inner core encapsulated with a layer of dielectric liquid (e.g., oil or lipid layer), which is immiscible to the bulk aqueous medium. The peripheral liquid layer of the undertaken nanoparticle contains mobile charges due to presence of solubilized surfactants. The mobile electrolyte ions can penetrate across the peripheral layer depending on the difference in the Born energy of the both phases. Such types of nanoparticles have received substantial attention due to their widespread applications in biomedical research. The electric double layer (EDL) is governed by the linearized Poisson-Boltzmann equation under a low potential limit and the electroosmotic flow field is governed by modified Stokes equation. We adopt the flat-plate formalism to obtain the closed analytical expression for the electrophoretic mobility of the undertaken particle under a thin EDL approximation. The dependence of electrophoretic mobility on the pertinent parameters is also illustrated.

Impact of ion-steric and ion-partitioning effects on electrophoresis of soft particles.

A theoretical study on the electrophoresis of a soft particle is made by taking into account the ion steric interactions and ion partitioning effects under a thin Debye layer consideration with negligible surface conduction. Objective of this study is to provide a simple expression for the mobility of a soft particle which accounts for the finite-ion-size effect and the ion partitioning arise due to the Born energy difference between two media. The Donnan potential in the soft layer is determined by considering the ion steric interactions and the ion partitioning effect. The volume exclusion due to the finite ion size is considered by the Carnahan-Starling equation and the ion partitioning is accounted through the difference in Born energy. The modified Poisson-Boltzmann equation coupled with Stokes-Darcy-Brinkman equations are considered to determine the mobility. A closed-form expression for the electrophoretic mobility is obtained, which reduces to several existing expressions for mobility under various limiting cases.

Electrokinetic ion transport and fluid flow in a pH-regulated polymer-grafted nanochannel filled with power-law fluid.

In this article, we have discussed extensively electrokinetic ion transport and fluid flow through a slit polymer-grafted nanochannel filled with power-law fluid. The rigid walls of the channel are coated with the ion and fluid penetrable polymer layer containing a pH-regulated zwitterionic functional group (e.g., succinoglycan). The mathematical model is based on the non-linear Poisson-Boltzmann equation for electric double layer potential and the flow field within the polymer layer is governed by a modified Darcy-Brinkman equation; the Cauchy momentum equation governs the fluid flow outside of the polymer layer along with the equation of continuity for incompressible fluid. In order to consider a wide range of pertinent parameters, we adopt a finite difference based numerical tool to solve the coupled set of governing equations. We have analyzed several interesting features of electrokinetic transport phenomena through such a polymer-grafted nanochannel for a wide range of electrostatic and hydrodynamic properties of the polymer layer, parameters describing the non-Newtonian rheology of the background fluid, and the pH and concentration of the bulk electrolyte. In addition, we have also illustrated the ionic current across the undertaken nanochannel and observed that it can be either cation selective, anion selective or non-selective, depending on the critical choice of the pertinent parameters.

Analytic Expression for Electrophoretic Mobility of Soft Particles with a Hydrophobic Inner Core at Different Electrostatic Conditions.

This paper presents a simplified model for the electrophoresis of a soft particle with a nonwettable rigid core with charged polyelectrolyte corona under a weak-field and low-charge density consideration. We have derived a closed form solution for the mobility, which reduces to the well-known expressions for mobility as derived by Ohshima for limiting cases such as a hydrophilic charged core coated with an uncharged polymer (Ohshima, H. J. Colloid Interface Sci. 2002, 252, 119-125) or an uncharged no-slip core coated with a polyelectrolyte layer (Ohshima, H. Electrophoresis 2006, 27, 526-533). The generalized mobility expression reduces to the existing expression for mobility of a rigid hydrophobic colloid as the soft layer shrinks to zero. The general form of the mobility expression involves elliptic integrals, which can be computed easily through a software like Mathematica. We have derived analytical solutions for mobility pertaining to several particular cases. The occurrence of mobility reversal when the core and polyelectrolyte layer has a charge of opposite polarity is demonstrated in this paper.

Electrophoresis of composite soft particles with differentiated core and shell permeabilities to ions and fluid flow.

Within the framework of analytical theories for soft surface electrophoresis, soft particles are classically defined by a hard impermeable core of given surface charge density surrounded by a polyelectrolyte shell layer permeable to both electroosmotic flow and ions from background electrolyte. This definition excludes practical core-shell particles, e.g. dendrimers, viruses or multi-layered polymeric particles, defined by a polyelectrolytic core where structural charges are distributed and where counter-ions concentration and electroosmotic flow velocity can be significant. Whereas a number of important approximate expressions has been derived for the electrophoretic mobility of hard and soft particles, none of them is applicable to such generic composite core-shell particles with differentiated ions- and fluid flow-permeabilities of their core and shell components. In this work, we elaborate an original closed-form electrophoretic mobility expression for this generic composite particle type within the Debye-Hückel electrostatic framework and thin double layer approximation. The expression explicitly involves the screening Debye layer thickness and the Brinkman core and shell hydrodynamic length scales, which favors so-far missing analysis of the respective core and shell contributions to overall particle mobility. Limits of this expression successfully reproduce results from Ohshima's electrophoresis theory solely applicable to soft particles with or without hard core.

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.

Ion partitioning effect on the electrophoresis of a soft particle with hydrophobic core.

A theoretical study on the electrophoresis of a soft particle made up of a charged hydrophobic inner core surrounded by polyelectrolyte layer (PEL) is made. The dielectric permittivity of the PEL and aqueous solution are considered to be different, which creates the ion partitioning effect. The ion partitioning effect, which is accounted by the Born energy difference, modifies the distribution of mobile ions in the PEL and hence alters the particle electrophoresis. The combined effects of core hydrophobicity and the ion partitioning effect on the mobility are determined based on the Debye-Huckel approximation under a thin Debye layer assumption. An analytic expression for the electrophoretic mobility taking into account the core hydrophobicity and ion partitioning effect is obtained. The occurrence of zero mobility and reversal of mobility of the soft particle is illustrated.