Electrophoresis of a sphere along the axis of a cylindrical pore: effects of double-layer polarization and electroosmotic flow

The electrophoresis of a rigid sphere along the axis of a cylindrical pore is investigated theoretically. Previous analysis is extended to the case where the effects of double-layer polarization and electroosmotic flow can be significant. The influences of the surface potential, the thickness of the double layer, and the relative size of a pore on the electrophoretic behavior of a sphere are discussed. Some interesting results are observed. For example, if both a sphere and a pore are positively charged, then the mobility of the sphere has a local minimum as the thickness of its double layer varies. Depending upon the level of the surface potential of a sphere and the degree of significance of the boundary effect, the mobility of the sphere may change its sign twice as the thickness of its double layer varies. This result can play a significant role in electrophoresis measurements.     

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.     

Effect of charged boundary on electrophoresis: Sphere in spherical cavity at arbitrary potential and double-layer thickness

The boundary effect on electrophoresis is investigated by considering a spherical particle at an arbitrary position in a spherical cavity. Our previous analysis is extended to the case where the effect of double-layer polarization can be significant. Also, the effect of a charged boundary, which yields an electroosmotic flow and a pressure gradient, thereby making the problem under consideration more complicated, is investigated. The influences of the level of the surface potential, the thickness of double layer, the relative size of a sphere, and its position in a cavity on the electrophoretic behavior of the sphere are discussed. Some results that are of practical significance are observed. For example, if a positively charged sphere is placed in an uncharged cavity, its mobility may have a local minimum as the thickness of the double layer varies. If an uncharged sphere is placed in a positively charged cavity, the mobility may have a local minimum as the position of the sphere varies. Also, if the size of a sphere is fixed, its mobility may have a local minimum as the size of a cavity varies. These provide useful information for the design of an electrophoresis apparatus.     

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

The electrophoresis of a rigid, charge-regulated, spherical particle normal to a large disk is investigated under the conditions of low surface potential and weak applied electric field. We show that, although the presence of a charged disk does not generate an electroosmotic flow, it affects particle motion appreciably through inducing charge on its surface and establishing an osmotic pressure field. The competition between the hydrodynamic force and the electric force may yields a local extremum in mobility; it is also possible that the direction of particle movement is reversed. In general, if a particle remains at constant surface potential, a decrease in the thickness of double layer has the effect of increasing the electrostatic force acting on it so that its mobility increases. However, this might not be the case for a charged-regulated particle because an excess hydrodynamic force is enhanced. For a fixed separation distance, the influence of a charged disk on mobility may reduce to a minimum if the bulk concentration of hydrogen ion is equal to the dissociation constant of the monoprotic acidic functional groups on particle surface.     

Sedimentation of a composite particle in a spherical cavity

The boundary effect on the sedimentation of a colloidal particle is investigated theoretically by considering a composite sphere, which comprises a rigid core and an ion-penetrable membrane layer, in a spherical cavity. A pseudo-spectral method is adopted to solve the governing electrokinetic equations, and the influences of the key parameters on the sedimentation behavior of a particle are discussed. We show that both the qualitative and quantitative behaviors of a particle are influenced significantly by the presence of the membrane layer. For example, if the membrane layer is either free of fixed charge or positively charged and the surface potential of the rigid core is sufficiently high, the sedimentation velocity has a local minimum and the sedimentation potential has a local maximum as the thickness of the double layer varies. These local extrema are not observed when the membrane layer is negatively charged. If the double layer is thin, the influence of the fixed charge in the membrane layer on the sedimentation potential is inappreciable.     

Electrophoresis of two identical rigid spheres in a charged cylindrical pore

The electrophoresis of two identical spheres moving along the axis of a long cylindrical pore under the conditions of low surface potential and weak applied electric field is investigated. The geometry considered allows us to examine simultaneously the effects of boundary and the presence of a nearby entity on the behavior of a particle. The influences of the separation distance between two spheres, the thickness of a double layer, the ratio (radius of sphere/radius of pore), and the charged conditions on the surfaces of the spheres and the pore on the mobility of a particle are investigated. Several interesting results that are not reported in the literature are observed. For instance, although for the case of two positively charged spheres in an uncharged pore the qualitative behavior of a sphere depends largely on its size relative to that of a pore and the thickness of the double layer, this might not be the case when two uncharged spheres are in a positively charged pore. In addition, in the latter, the mobility of a sphere increases with the increases in the separation distance between two spheres, and this effect is pronounced when the ratio (radius of sphere/radius of pore) takes a medium value or the thickness of the double layer is either sufficiently thin or sufficiently thick.     

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).     

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.     

Boundary effect on electrophoresis: finite cylinder in a cylindrical pore

The boundary effect on electrophoresis is investigated by considering a finite cylindrical particle moving along the axis of a long cylindrical pore under conditions of low surface potential and weak applied electric field. The influence of the thickness of the double layer, the aspect ratio of a particle, the ratio particle radius/pore radius, and the charged conditions of the surfaces of the particle and pore on the electrophoretic behavior of a particle are investigated. We show that the effect of the aspect ratio of a particle on its electrophoretic behavior for the case where the particle is charged and the pore is uncharged is larger than that for the case where the particle is uncharged and the pore is charged. Also, depending on the parameters chosen, increasing the aspect ratio of a particle can either promote or hinder its movement, which is not reported in previous studies, and can play a role in electrophoresis measurements. Because both the electric and the flow fields in the gap between the particle and the pore are mediated by those near the top and the end of the particle, the end effect is large when the double layer is thick.     

Diffusiophoresis of a nonuniformly charged sphere in an electrolyte solution

The diffusiophoresis of a rigid, nonuniformly charged spherical particle in an electrolyte solution is analyzed theoretically focusing on the influences of the thickness of double layer, the surface charge distribution, the effect of electrophoresis, and the effect of double-layer polarization. We show that the nonuniform charge distribution on the particle surface yields complicated effect of double-layer polarization, leading to interesting diffusiophoretic behaviors. For example, if the sign of the middle part of the particle is different from that of its left- and right-hand parts, then depending upon the charge density and the fraction of the middle part, the particle can move either to the high-concentration side or to the low-concentration side. Both the diffusiophoretic velocity and its direction can be manipulated by the distribution of the surface charge density. In particular, if the electrophoresis effect is significant, then those properties are governed by the averaged surface charge density of the particle. A dipolelike particle, where its left- (right-) hand half is negatively (positively) charged, always migrates toward the low-concentration (left-hand) side, that is, it has a negative diffusiophoretic velocity. In addition, that diffusiophoretic velocity has a negative local minimum as the thickness of double layer varies.     

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 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.     

Sedimentation of a concentrated dispersion of composite colloidal particles

The sedimentation of a concentrated spherical dispersion of composite particles, where a particle comprises a rigid core and a membrane layer containing fixed charge, is investigated theoretically. The dispersion is simulated by a unit cell model, and a pseudo-spectral method based on Chebyshev polynomials is adopted to solve the problem numerically. The influences of the thickness of double layer, the concentration of particles, the surface potential of the rigid core of a particle, and the amount of fixed charge in the membrane layer on both the sedimentation potential and the sedimentation velocity are discussed. Several interesting results are observed; for example, depending upon the charged conditions on the rigid core and in the membrane layer of a particle, the sedimentation potential might have both a local maximum and a local minimum and the sedimentation velocity can have a local minimum as the thickness of double layer varies. Also, the sedimentation velocity can have a local maximum as the surface potential varies. We show that the sedimentation potential increases with the concentration of particles. The relation between the sedimentation velocity and the concentration of particles, however, depends upon the thickness of double layer.     

Electrophoresis of a membrane-coated sphere in a spherical cavity

The boundary effect on the electrophoresis of particles covered by a membrane layer is discussed by considering a spherical particle in a spherical cavity under the conditions where the effect of double-layer polarization can be significant. The influence of the key parameters of the system under consideration on the electrophoretic mobility of a particle is investigated. These include the surface potential; the thickness of the double layer; the relative size of the cavity; and the thickness, the fixed charge density, and the friction coefficient of the membrane layer. The fixed charge in the membrane layer of a particle is found to have a significant influence on its electrophoretic behavior. For instance, depending upon the amount of fixed charge in the membrane layer, the mobility of a particle may exhibit a local minimum as the thickness of the double layer varies.     

Electrophoresis of a particle at an arbitrary surface potential and double layer thickness: importance of nonuniformly charged conditions

Recent advances in material science and technology yield not only various kinds of nano- and sub-micro-scaled particles but also particles of various charged conditions such as Janus particles. The characterization of these particles can be challenging because conventional electrophoresis theory is usually based on drastic assumptions that are unable to realistically describe the actual situation. In this study, the influence of the nonuniform charged conditions on the surface of a particle at an arbitrary level of surface potential and double layer thickness on its electrophoretic behavior is investigated for the first time in the literature taking account of the effect of double-layer polarization. Several important results are observed. For instance, for the same averaged surface potential, the mobility of a nonuniformly charged particle is generally smaller than that of a uniformly charged particle, and the difference between the two depends upon the thickness of double layer. This implies that using the conventional electrophoresis theory may result in appreciable deviation, which can be on the order of ca. 20%. In addition, the nonuniform surface charge can yield double vortex in the vicinity of a particle by breaking the symmetric of the flow field, which has potential applications in mixing and/or regulating the medium confined in a submicrometer-sized space, where conventional mixing devices are inapplicable.     

Effects of double-layer polarization and electroosmotic flow on the electrophoresis of an ellipsoid in a spherical cavity

The electrophoresis of a rigid, positively charged ellipsoidal particle at the center of a spherical cavity is investigated theoretically under the conditions where the effects of double-layer polarization and the presence of an electroosmotic flow can be important. The equations governing the problem under consideration and the associated boundary conditions are solved numerically, and the influences of the key parameters on the electrophoretic mobility of the particle are discussed. We show that if the cavity is uncharged, the effect of double-layer polarization yields a local minimum in the electrophoretic mobility as the thickness of the double layer varies. This local minimum disappears if the cavity is also positively charged. In addition to reducing the scaled mobility of an ellipsoid, the presence of the boundary is also capable of influencing the relative magnitudes of the scaled mobility for particles of various shapes. For instance, if the volume of an ellipsoid is fixed, the scaled mobility ranks as prolate > sphere > oblate if the boundary effect is unimportant, but that order is reversed if the boundary effect is important.     

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.     

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.     

Electrophoresis of a toroid along the axis of a cylindrical pore

The electrophoresis of a toroid (doughnut-shaped entity) along the axis of a long cylindrical pore is analyzed under the conditions of low surface potential and weak applied electric field. The system under consideration is capable of modeling the electrophoretic behavior of various types of biocolloid such as bacterial DNA, plasmid DNA, and anabaenopsis, in a confined space. The influences of the key parameters of the problem, including the sizes of a toroid, the radius of a pore, and the thickness of the double layer, on the electrophoretic mobility of a toroid are discussed. We show that the electrophoretic behavior of a toroid under typical conditions can be different from that of an integrated entity. For instance, although the presence of the pore wall has the effect of retarding the movement of a particle, it becomes advantageous if a toroid is sufficiently close to the boundary. Several interesting behaviors are also observed, for example, the mobility of a toroid when the boundary effect is significant can be larger than that when it is insignificant.