Effect of a charged boundary on electrophoresis: a sphere at an arbitrary position in a spherical cavity

The effect of the presence of a charged boundary on the electrophoretic behavior of a particle is investigated by considering a sphere at an arbitrary position in a spherical cavity under conditions of low surface potential and weak applied electric field. Previous analyses are modified by using a more realistic electrostatic force formula and several interesting results, which are not reported in the literature, are observed. We show that the qualitative behavior of a particle depends largely on its position, its size relative to that of a cavity, and the thickness of the electric double layer. In general, the presence of a cavity has the effect of increasing the conventional hydrodynamic drag on a particle through a nonslip condition on the former. Also, a decrease in the thickness of the double layer surrounding a sphere has the effect of increasing the electrostatic force acting on its surface so that its mobility increases. However, this may not be the case when an uncharged particle in placed in a positively charged cavity, where the electroosmotic flow plays a role; for example, the mobility can exhibit a local maximum and the direction of electrophoresis can change.     

Electrophoresis of a rigid sphere in a Carreau fluid normal to a large charged disk

The electrophoresis of a rigid sphere in a Carreau fluid normal to a large disk is analyzed theoretically under the conditions of low surface potential and weak applied electric field. Previous analyses are extended to the case where a disk can be charged, and a more realistic electrostatic force formula is applied. We show that the qualitative behavior of a sphere depends largely on its distance from a disk, the thickness of double layer, and the nature of a fluid. In general, the presence of a disk has the effect of increasing the conventional hydrodynamic drag on a sphere, and a decrease in the thickness of the double layer surrounding a sphere has the effect of enhancing the shear-thinning effect. However, this might not be the case if a sphere is uncharged and a disk is charged, where the osmotic pressure field and the induced charge on the sphere surface can be significant. The shear-thinning effect is important only if the thickness of double layer is sufficiently thick. This result can play a significant role in practice such as in electrophoretic deposition, where the deposition electrode is charged and the fluid medium is usually of shearing-thinning nature.     

Electrophoresis of a charge-regulated sphere at an arbitrary position in a charged spherical cavity

The electrophoresis of a charge-regulated spherical particle at an arbitrary position in a charged spherical cavity is modeled under conditions of low surface potential (<25 mV) and weak applied electric field (<25 kV/m). The charged cavity allows us to simulate the effect of electroosmotic flow, and the charge-regulated nature of the particle permits us to model various types of surface. The problem studied previously is reanalyzed based on a more rigorous electric force formula. In particular, the influences of various types of charged conditions on the electrophoretic behavior of a particle and the roles of all the relevant forces acting on the particle are examined in detail. Several new results are found. For instance, the mobility of a particle has a local minimum as the thickness of a double layer varies, which is not seen in the cases where the surface of a particle is maintained at a constant potential and at a constant charge density.     

Electrophoresis of a charge-regulated toroid normal to a large disk

The electrophoresis of a charge-regulated toroid (doughnut-shaped entity) normal to a large disk is investigated under the conditions of low surface potential and weak applied electric field. The system considered is capable of modeling the electrophoretic behavior of various types of biocolloids such as bacterial DNA, plasmid DNA, and anabaenopsis near a perfectly conducting planar wall. The influences of the size of the toroid, the separation distance between the toroid and the disk, the charged conditions on the surfaces of the toroid and the disk, and the thickness of electric double layer on the electrophoretic mobility of the toroid are discussed. The results of numerical simulation reveal that under typical conditions the electrophoretic behavior of the toroid can be different from that of an integrated entity. For instance, if the surface of the toroid carries both acidic and basic functional groups, its mobility may have a local maximum as the thickness of double layer varies. We show that the electrophoretic behavior of the toroid is different, both qualitatively and quantitatively, from that of the corresponding integrated particle (particle without hole).     

Electrophoretic motion of a spherical particle in a converging-diverging nanotube

A charged spherical particle is concentrically positioned in a converging-diverging nanotube filled with an electrolyte solution, resulting in an electric double layer (EDL) forming around the particle's surface. In the presence of an axially applied electric field, the particle electrophoretically migrates along the axis of the nanotube due to the electrostatic and hydrodynamic forces acting on the particle. In contrast to a cylindrical nanotube with a constant cross-sectional area in which the electric field is almost uniform, the presence of a converging-diverging section in a nanotube alters the electric field, perturbs the charge distribution, and induces a pressure gradient and a recirculating flow that affect the electrostatic and hydrodynamic forces acting on both the particle and the fluid. Depending on the magnitude of the surface charge density along the nanotube's wall, the particle's electrophoretic motion may be significantly accelerated as the particle transverses the converging-diverging section. A continuum model consisting of the Nernst-Planck, Poisson, and Navier-Stokes equations for the ionic concentrations, electric potential, and flow field is implemented to compute the particle's velocity as a function of the particle's size, the nanotube's geometry, surface charges, electric field intensity, bulk electrolyte concentration, and the particle's location. When the particle is negatively charged and the wall of the nanotube is uncharged, the particle migrates in the direction opposite to that of the applied electric field and the presence of the converging-diverging section significantly accelerates the particle's motion. This, however, is not always true when the nanotube's wall is charged with the same sign as that of the particle. Once the ratio of the surface charge density of the nanotube's wall to that of the particle exceeds a certain value, the negatively charged particle will not translocate through the tube toward the anode and does not attain the maximum velocity at the throat of the converging-diverging section. One can envision such a device to be a nanofilter that allows molecules with surface charge densities much higher than that of the wall to go through the nanofilter, while preventing molecules with surface charge densities lower than that of the wall from passing through the nanofilter. The induced recirculating flow may be used to enhance mixing and to stretch, fold, and trap molecules in nanofluidic detectors and reactors.     

Electrophoresis of a sphere at an arbitrary position in a spherical cavity filled with Carreau fluid

Boundary effects on the electrophoretic behavior of a charged entity are of both fundamental and practical significance. Here, they are examined by considering the case where a sphere is at an arbitrary position in a spherical cavity under conditions of low surface potential and weak applied electrical field. Previous analyses are extended to the case of a non-Newtonian fluid, and a Carreau model is adopted for this purpose. The effects of key parameters such as the thickness of a double layer, the relative sizes of particle and cavity, the position of a particle, and the nature of a fluid on the electrophoretic mobility of a particle are discussed. Several interesting phenomena are observed. For example, if the applied electric field points toward north, the mobility of a particle has a local maximum when it is at the center of a cavity. However, if a particle is sufficiently close to the north pole of a cavity, its mobility exhibits a local minimum as its position varies. This does not occur when the particle is close to the south pole of the cavity; instead, it may move in the direction opposite to that of the applied electric field. For a Newtonian fluid, if a particle is close to the north pole of a cavity, its upward movement yields a clockwise (counterclockwise) vortex near the north pole of the cavity and a counterclockwise (clockwise) vortex near the south pole of the cavity on its right (left)-hand side. The latter is not observed for a Carreau fluid.     

Electrophoresis of a finite cylinder positioned eccentrically along the axis of a long cylindrical pore

The electrophoresis of a finite cylindrical particle positioned eccentrically along the axis of a long cylindrical pore is modeled under the conditions of low surface potential and weak applied electric field. The influences of the eccentricity of a particle and its linear size, the radius of the pore, and the thickness of the electrical double layer on the electrophoretic mobility of the particle are investigated. Some interesting results are observed. For instance, for the case of a positively charged particle placed in an uncharged pore, if the double layer is thin and the particle is short, the mobility has a local minimum as the eccentricity varies. Also, for a short particle the mobility at a thinner double layer can be smaller than that at a thicker double layer, which has never been reported for the case of constant surface potential. In general, the mobility increases with the increase in the eccentricity, and this effect is pronounced when the size of a particle is large and/or the radius of a pore is small.     

Transient electrophoresis in a suspension of charged particles with arbitrary electric double layers

The startup of electrophoretic motion in a suspension of spherical colloidal particles, which may be charged with constant zeta potential or constant surface charge density, due to the sudden application of an electric field is analytically examined. The unsteady modified Stokes equation governing the fluid velocity field is solved with unit cell models. Explicit formulas for the transient electrophoretic velocity of the particle in a cell in the Laplace transforms are obtained as functions of relevant parameters. The transient electrophoretic mobility is a monotonic decreasing function of the particle-to-fluid density ratio and in general a decreasing function of the particle volume fraction, but it increases and decreases with a raise in the ratio of the particle radius to the Debye length for the particles with constant zeta potential and constant surface charge density, respectively. On the other hand, the relaxation time in the growth of the electrophoretic mobility increases substantially with an increase in the particle-to-fluid density ratio and with a decrease in the particle volume fraction but is not a sensitive function of the ratio of the particle radius to the Debye length. For specified values of the particle volume fraction and particle-to-fluid density ratio in a suspension, the relaxation times in the growth of the particle mobility in transient electrophoresis and transient sedimentation are equivalent.     

Dynamic Mobility of Two Spherical Particles with Thick Double Layers

The dynamic electrophoretic mobility of a pair of nearby spherical particles is analyzed in the case when the thickness of the electrical double layer around each particle is comparable to the particle radius. By means of an integral reciprocal relation, a formal expression is obtained for the force and torque on N spheres subject to an oscillating electric field which may be spatially varying. Upon linearizing in the surface potential, this expression is shown to depend upon a set of purely hydrodynamic problems involving N neutral spheres, the calculation of the electric field around N neutral spheres, and the equilibrium charge distribution around N charged spheres. In the case of a single particle, the known analytic formula for the dynamic mobility is recovered. For a pair of identical particles, the dynamic mobility is calculated numerically, using known solutions to the required subproblems. An analytical expression for the mobility of a pair of widely separated spheres is also obtained by a method of reflections, and this is in excellent agreement with the numerical results outside the range of double layer overlap. Copyright 2000 Academic Press.     

Electrokinetic boundary condition compatible with the Onsager reciprocal relation in the thin double layer approximation

The boundary condition, which has been used in the conventional electrokinetic calculation in the thin double layer approximation, has a flaw that it does not give the Onsager reciprocal relation for the sedimentation of charged particle. We propose a new boundary condition, which satisfies the reciprocal relation, and derive a general form for the mobility matrix for the motion of a charged particle under the action of external force, torque, and electric field. We then calculate the mobility matrix explicitly for homogeneously charged spherical particle and discuss the effect of the surface slippage and the surface conductivity on the particle mobility and electric conductivity.     

Dynamic electrophoresis of a sphere in a spherical cavity: arbitrary surface potential

The boundary effect on the dynamic electrophoretic behavior of a charged entity is examined by considering a sphere in a spherical cavity. The present study extends previous analysis to the case of an arbitrary level of electrical potential where the effect of double-layer distortion can be significant. The governing equations are solved numerically based on a pseudo-spectral method, which is found to be sufficient in solving the corresponding electrophoresis problem when a static electric field is applied. The result of numerical simulation reveals that as the size of a cavity decreases, both the magnitude of the mobility and the inertial force acting on a particle decrease accordingly. Also, while the distortion of the ionic cloud should not be ignored, in general, when the surface potential of a particle is high, its influence on the magnitude and on the phase angle of the mobility is alleviated by the presence of the cavity.     

On the limiting electrophoretic mobility of a highly charged colloidal particle in an electrolyte solution

The electrophoretic mobility of a spherical charged colloidal particle in an electrolyte solution with large kappaa (where kappa= Debye-Hückel parameter and a= particle radius) tends to a nonzero constant value in the limit of high zeta potential. It is demonstrated that this is caused by the fact that counterions condensed near the highly charged particle surface do not contribute to the electrophoretic mobility and only co-ions govern the mobility. A simple method to derive the limiting electrophoretic mobility expression is given. The present method is also applied to cylindrical particles, showing that the leading term of the limiting electrophoretic mobility of a cylindrical particle in a transverse field with large kappaa is the same as that of a spherical particle. The electrophoretic mobility of a cylindrical particle in a tangential field, on the other hand, is proportional to the particle zeta potential and does not exhibit a constant limiting value for high zeta potentials.     

Sedimentation of a charged colloidal sphere in a charged cavity

An analytical study is presented for the quasisteady sedimentation of a charged spherical particle located at the center of a charged spherical cavity. The overlap of the electric double layers is allowed, and the polarization (relaxation) effect in the double layers is considered. The electrokinetic equations that govern the ionic concentration distributions, electric potential profile, and fluid flow field in the electrolyte solution are linearized assuming that the system is only slightly distorted from equilibrium. Using a perturbation method, these linearized equations are solved for a symmetric electrolyte with the surface charge densities of the particle and cavity as the small perturbation parameters. An analytical expression for the settling velocity of the charged sphere is obtained from a balance among the gravitational, electrostatic, and hydrodynamic forces acting on it. Our results indicate that the presence of the particle charge reduces the magnitude of the sedimentation velocity of the particle in an uncharged cavity and the presence of the fixed charge at the cavity surface increases the magnitude of the sedimentation velocity of an uncharged particle in a charged cavity. For the case of a charged sphere settling in a charged cavity with equivalent surface charge densities, the net effect of the fixed charges will increase the sedimentation velocity of the particle. For the case of a charged sphere settling in a charged cavity with their surface charge densities in opposite signs, the net effect of the fixed charges in general reduces/increases the sedimentation velocity of the particle if the surface charge density of the particle has a greater/smaller magnitude than that of the cavity. The effect of the surface charge at the cavity wall on the sedimentation of a colloidal particle is found to increase with a decrease in the particle-to-cavity size ratio and can be significant in appropriate situations.     

Communication: The phoretic drift of a charged particle animated by a direct ionic current

A charged colloidal particle which is suspended in an electrolyte solution drifts due to an external voltage application. For direct currents, particle motion is affected by two separate mechanisms: electro-osmotic slip associated with the electric field and chemi-osmotic slip associated with the inherent salt concentration gradient in the solution. These two mechanisms are interrelated and are of comparable magnitude. Their combined effect is demonstrated for cation-exchange electrodes using a weak-current approximation. The linkage between the two mechanisms results in an effectively modified mobility, whose dependence on the particle zeta potential is nonlinear. At small potentials, the electro-osmotic mechanism dominates and the particle migrates according to the familiar Smoluchowski mobility, linear in the electric field. At large zeta potentials, chemiosmosis becomes dominant: for positively charged particles, it tends to arrest motion, leading to mobility saturation; for negatively charged particles, it enhances the drift, effectively leading to a shifted linear dependence of the mobility on the zeta potential, with twice the Smoluchowski slope.     

Electrophoretic Motion of a Spherical Particle with a Thick Double Layer in Bounded Flows

The electrophoretic mobility of a spherical colloidal particle with low zeta potential near a solid charged boundary is calculated numerically for arbitrary values of the double layer thickness by a generalization of Teubner's method to the case of bounded flow. Three examples are considered: a sphere near a nonconducting planar wall with electric field parallel to the wall, near a perfectly conducting planar wall with electric field perpendicular to the wall, and on the axis of a cylindrical pore with electric field parallel to the axis. The results are compared with recent analytical calculations using the method of reflections. For the case of a charged sphere near a neutral surface, the reflection results are quite good, provided there is no double layer overlap, in which case there can be extra effects for constant potential particles that are entirely missed by the analytical expressions. For a neutral sphere near a charged surface, the reflection results are less successful. The main reason is that the particle feels the profile of the electroosmotic flow, an effect ignored by construction in the method of reflections. The general case is a combination of these, so that the reflections are more reliable when the electrophoretic motion dominates the electroosmotic flow. The effect on particle mobility of particle-wall interactions follows the trend expected on geometric grounds in that sphere-plane interactions are stronger than sphere-sphere interactions and the effect on a sphere in a cylindrical pore is stronger still. In the latter case, particle mobility can fall by more than 50% for thick double layers and a sphere half the diameter of the pore. The agreement between numerical results and analytical results follows the same trend, being worst for the sphere in a pore. Nevertheless, the reflections can be reliable for some geometries if there is no double layer overlap. This is demonstrated for a specific example where reflection results have previously been compared with experiments on protein mobility through a membrane (J. Ennis et al., 1996, J. Membrane Sci. 119, 47). Copyright 1999 Academic Press.     

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.     

Electrophoresis of a charge-regulated spheroid along the axis of an uncharged cylindrical pore

Boundary effects can have a profound influence on the electrophoretic behavior of a charged entity, in particular, when the entity is nonspherical and its surface conditions are dependent upon the nearby environment. In this study, the electrophoresis of a spheroid along the axis of an uncharged cylindrical pore is analyzed for the case where the electrical potential is low and the applied electric field is weak. We consider the case where the surface of a particle contains dissociable acidic and basic functional groups, which simulate biological colloids and entities covered by an artificial membrane. This leads to a mixed-type boundary value problem, which extends the conventional constant-surface-potential and constant-surface-charge-density models to a more general case. The effects of the particle aspect ratio, the relative magnitudes of particle and pore, the thickness of the double layer surrounding a particle, and the pH of the liquid phase on the electrophoretic mobility of a particle are investigated. Several interesting results are observed; for example, if the volume of a particle is fixed, its mobility may have a local maximum as the relative magnitudes of its two axes vary.     

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.     

Electrophoresis of a spherical particle along the axis of a cylindrical pore: effect of electroosmotic flow

The electrophoresis of a spherical particle along the axis of a cylindrical pore is investigated under conditions of low surface potential and thick double layer. In particular, the effect of electroosmotic flow is taken into account. The results of numerical simulation reveal that if both particle and pore are positively charged, the variation of the mobility of a particle may have a local minimum as the thickness of the double layer varies, which is not reported in the literature. This is mainly due to the charge induced on the particle surface, which arises from the presence of the charged boundary. Depending upon the level of the surface potential of the pore, the presence of the local minima may lead to a reversal in the direction of particle movement as the thickness of the double layer surrounding it varies: if the surface potential is either too low or too high, reversal does not occur; if it has a medium level, reversal occurs twice. This interesting observation can play a role in electrophoresis measurements. Previous analysis predicts that reversal always occurs once, regardless of the level of the surface potential of the pore.