Electrophoretic Mobility of a Sphere in a Spherical Cavity

Electrophoresis of a dielectric droplet with constant surface charge density

Electrophoresis of a dielectric fluid droplet with constant surface charge density is investigated theoretically in this study. A pseudo-spectral method based on Chebyshev polynomials is adopted to solve the governing electrokinetic equations. It is found, among other things, that the larger the electrolyte strength in the ambient solution is, the slower the droplet moves in general. This is due to the strong screening effect of the large amount of indifferent counterions in the neighborhood of the droplet, with no reinforcement of potential-determining ions adsorbing to the droplet surface. The droplet comes to a complete halt eventually. Critical points are discovered for highly charged droplets, at which the droplet surface becomes immobile and the interior fluid stops recirculating. The droplet moves like a rigid particle with constant mobility regardless of its viscosity, a situation referred to as the "solidification phenomenon." The deadlock between the spinning motions on the charged droplet surface induced by the electric driving force and the hydrodynamic driving force respectively is responsible for this peculiar phenomenon. This is also observed for a dielectric droplet with constant surface electric potential. We demonstrate here that it occurs in the constant surface charge density situation as well.

Frequency-controlled electrophoretic mobility of a particle within a porous, hollow shell

The unique properties of yolk-shell or rattle-type particles make them promising candidates for applications ranging from switchable photonic crystals, to catalysts, to sensors. To realize many of these applications it is important to gain control over the dynamics of the core particle independently of the shell.

HYPOTHESIS

The core particle may be manipulated by an AC electric field with rich frequency-dependent behavior.

EXPERIMENTS

Here, we explore the frequency-dependent dynamic electrophoretic mobility of a charged core particle within a charged, porous shell in AC electric fields both experimentally using liquid-phase electron microscopy and numerically via the finite-element method. These calculations solve the Poisson-Nernst-Planck-Stokes equations, where the core particle moves according to the hydrodynamic and electric forces acting on it.

FINDINGS

In experiments the core exhibited three frequency-dependent regimes of field-driven motion: (i) parallel to the field, (ii) diffusive in a plane orthogonal to the field, and (iii) unbiased random motion. The transitions between the three observed regimes can be explained by the level of matching between the time required to establish ionic gradients in the shell and the period of the AC field. We further investigated the effect of shell porosity, ionic strength, and inner-shell radius. The former strongly impacted the core's behavior by attenuating the field inside the shell. Our results provide physical understanding on how the behavior of yolk-shell particles may be tuned, thereby enhancing their potential for use as building blocks for switchable photonic crystals.

Diffusiophoresis of a Highly Charged Porous Particle Induced by Diffusion Potential

Diffusiophoresis, the motion of a colloidal particle in response to the concentration gradient of solutes in the suspending medium, is investigated theoretically on the basis of numerical computations in this study for charged porous particles, especially highly or extremely porous ones, focusing on the electrophoresis component induced by diffusion potential, which is generated spontaneously in a binary electrolyte solution where the diffusivities of the two ionic species are distinct. A benchmark carbonic acid solution of H_(aq)⁺ and HCO_3(aq)^- is chosen to be the major suspending medium, as its large diffusion potential and remarkable performance in practical applications have been reported recently in the literature. More than 3 orders of magnitude increase in particle diffusiophoretic mobility is predicted under some circumstances, should the permeability of the particle increase 10-fold. Nonlinear effects such as the motion-deterring double-layer polarization effect pertinent to highly charged particles and the counterion condensation or shielding/screening effect pertinent to porous particles are investigated in particular for their impact on the particle motion, among other electrokinetic parameters examined. A visual demonstration of the nonlinear double-layer polarization is provided. Moreover, both the chemiphoresis and the electrophoresis components are explored and analyzed in detail. The results presented here can be applied in biochemical and biomedical fields involving DNAs and proteins, which can be modeled excellently as charged porous particles in their electrokinetic motion.

Simulation of polyelectrolyte electrophoresis: effects of the aspect ratio, double-layer polarization, effective charge, and electroosmotic flow

The electrophoresis of a deformable polyelectrolyte (PE) is studied theoretically by considering a Poisson-Nernst-Planck model coupled with modified Navier-Stokes equations, taking account of the effects of double-layer polarization, counterion condensation, and electroosmotic flow. The influences of the local electric field and the effective PE charge on the PE mobility are discussed, thereby providing a complete picture for the phenomenon under consideration. Our model explains successfully the presence of a local minimum in the mobility of a highly charged PE as the bulk salt concentration varies, as observed experimentally. Numerical simulation also reveals several interesting and important results. For example, the more a PE is stretched in the direction of electrophoresis, the larger is its mobility. As the double layer becomes thin, the local electric field becomes independent of the PE shape, and its behavior mainly depends upon its effective charge. We show that the force that stretches a PE is maximal when it is spherical and decreases with an increasing aspect ratio, which has not been reported previously.

Electrokinetic behavior of a pH-regulated, zwitterionic nanocylinder in a cylindrical nanopore filled with multiple ionic species

Recent advances in fabrication techniques make nano-sized pores as promising platforms for both detection and sequencing of individual biopolymers such as DNA. To simulate the electrokinetic behavior of a particle in this case, we consider the electrophoresis of a soft nanocylinder comprising a rigid core and a pH-regulated, zwitterionic polyelectrolyte layer along the axis of a rigid cylindrical nanopore. Extending the conventional electrophoresis analysis, where the liquid phase contains only one kind each of cations and anions, we assume that it contains multiple ionic species, as is usually the case in practice. The key parameters are examined for their influences on the electrokinetic behavior of a particle. These include pH, the thickness of the polyelectrolyte layer and the density of its functional groups, and the pore size. The results gathered provide necessary information for the design of an electrokinetic apparatus.

Effect of multiple ionic species on the electrophoretic behavior of a charge-regulated particle

It is often assumed in the conventional electrophoresis analysis that the liquid phase contains only one kind of each cation and anion. That analysis is extended to the case where the liquid phase contains multiple ionic species in this study so that the conditions considered are closer to reality. Using a dispersion of SiO(2) particles, which is of a charge-regulated nature, as an example, where the dispersion pH is adjusted by HCl and NaOH, numerical simulation is conducted to examine the electrophoretic behaviors of the particle under various conditions. We show that the presence of multiple ionic species is capable of yielding profound and interesting electrophoretic behaviors, which are justified by the experimental data in the literature. In addition, we show that two types of double-layer polarization (DLP) are present that have not been reported previously in the electrophoresis analyses. Type I DLP, which reduces the mobility of a particle, occurs inside the double layer, and type II DLP, which raises that mobility, occurs immediately outside the double layer.

Diffusiophoresis of a soft sphere normal to two parallel disks

The diffusiophoresis of a soft spherical particle normal to two parallel disks subject to an applied ionic concentration gradient is modeled theoretically. The soft particle, which comprises a rigid core and a porous membrane layer, is capable of simulating a wide class of particles such as biocolloids and particles covered by an artificial membrane layer; a rigid particle can also be recovered as the limiting case where the membrane layer is infinitely thin. The problem considered simulates, for example, the chemotaxis of cells or microorganisms. We show that the presence of the membrane layer is capable of yielding complicated diffusiophoretic behavior when the sign of the charge carried by that layer is different from that on the surface of the rigid core of the particle. Both the sign and the magnitude of the diffusiophoretic velocity of a particle can be adjusted through varying the friction coefficient of its membrane layer. These results are of practical significance, for example, in the case where diffusiophoresis is adopted as a separation operation or as a tool to carry and/or control the rate of drug release.

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.

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

The influence of a charged boundary on the electrophoretic behavior of an entity in a non-Newtonian fluid is studied by considering a sphere at an arbitrary position in a spherical cavity filled with a Carreau fluid under the conditions of low surface potential and weak applied electric field. The dependence of the mobility of a sphere on its position in a cavity, the size of a cavity, the thickness of a double layer, and the nature of a fluid is investigated. In addition to the fact that the effect of shear-thinning is advantageous to the movement of a sphere, several other interesting results are also observed. For instance, if an uncharged sphere is in a positively charged cavity, where the electroosmotic flow and the induced charge on the sphere surface play a role, the effect of shear-thinning is important only if the thickness of the double layer is either sufficiently thin or sufficiently thick. However, this might not be the case if a positively charged sphere is in an uncharged cavity.

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 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.

Dynamic electrophoresis of droplet dispersions at low surface potentials

The dynamic electrophoresis of a dispersion of spherical droplets under conditions of low surface potential and arbitrary double-layer thickness and droplet volume fraction is analyzed. A cell model with the Shilov-Zharkikh boundary condition for the electric potential is adopted to simulate a dispersion, and the governing equations and the associated boundary conditions are solved by a pseudo-spectral method based on Chebyshev polynomials. The influence of the frequency of the applied electric field, the volume fraction of the droplets, the thickness of the double layer, and the relative magnitude of the viscosity of the droplet fluid on the electrophoretic behavior of a dispersion is discussed.

Evaluation of the electric force in electrophoresis

A new expression for the evaluation of the electric force acting on a colloidal particle in an applied electric field is derived under the condition of weak applied electric field. The expression derived, which is based on the Maxwell stress tensor, is applicable to both rigid and soft particles for various types of surface conditions and to both symmetric and asymmetric geometries. We show that, depending upon the electrophoresis conditions, the electric force evaluated by the methods commonly used in the literature can be overestimated, thereby leading to incorrect electrophoretic mobility.

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.

Dynamic electrophoresis of a droplet in a spherical cavity

The electrophoretic behavior of a droplet in a spherical cavity subject to an alternating electric field is analyzed theoretically under the conditions of an arbitrary level of surface potential and double-layer thickness. The influences of the thickness of the double layer, the level of surface potential, the size of a droplet, the viscosity of the droplet fluid, and the frequency of the applied electric field on the electrophoretic behavior of a droplet are examined through numerical simulations. We show that, because of the effect of double-layer deformation, the magnitude of the electrophoretic mobility of a droplet could have a local maximum and the phase angle could have a negative (phase lead) local minimum as the frequency of the applied electric field varies. In general, the lower the surface potential, the thicker the double layer and the larger the viscosity of the droplet fluid, and the more significant the boundary effect, the smaller the magnitude of the electrophoretic mobility of a droplet.

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 Spherical Dispersion of Polyelectrolytes in a Salt-Free Solution

The electrophoretic behavior of a spherical dispersion of polyelectrolytes of arbitrary concentration is analyzed theoretically under a salt-free condition, that is, the liquid phase contains only counterions which come from the dissociation of the functional groups of polyelectrolytes. We show that, in general, the surface potential of a polyelectrolyte increases nonlinearly with its surface charge. A linear relation exists between them, however, when the latter is sufficiently small; and the more dilute the concentration of polyelectrolytes, the broader the range in which they are linearly correlated. If the amount of surface charge is sufficiently large, counterion condensation occurs, and the rate of increase of surface potential as the amount of surface charge increases declined. Also, it leads to an inverse in the perturbed potential near the surface of a polyelectrolyte, and its mobility decreases accordingly. For a fixed amount of surface charge, the lower the concentration of polyelectrolytes and/or the lower the valence of counterions, the higher the surface potential. The qualitative behavior of the mobility of a polyelectrolyte as the amount of its surface charge varies is similar to that of its surface charge.

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.

Electrophoresis of a rigid sphere in a Carreau fluid normal to a planar surface

The boundary effect on electrophoresis is investigated by considering the electrophoresis of a spherical particle in a non-Newtonian fluid normal to a planar surface under conditions of low surface potential and weak applied electric field. The Carreau model, which is widely used for the description of polymeric fluids of shear-thinning nature, is adopted to simulate the non-Newtonian behavior of the fluid. We show that, in general, shear thinning has the effect of raising the electrophoretic mobility of a particle. The thinner the double layer, the more significant this effect is, and, since the presence of the planar surface has the effect of enhancing the shear-thinning effect, the closer a particle is to the planar surface, the larger is its mobility. Both the shear rate and the viscosity of the fluid vary most significantly in the gap between the particle and the planar surface, and the maximal shear rate and the minimal viscosity occur on the particle surface.

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.

Electrophoresis of a concentrated aqueous dispersion of non-Newtonian drops

The electrophoresis of a concentrated dispersion of non-Newtonian drops in an aqueous medium, which has not been investigated theoretically in the literature, is analyzed under conditions of low zeta potential and weak applied electric field. The results obtained provide a theoretical basis for the characterization of the nature of an emulsion and a microemulsion system. A Carreau fluid, which has wide applications in practice, is chosen for the non-Newtonian drops, and the unit cell model of Kuwabara is adopted to simulate a dispersion. The effects of the key parameters of a dispersion, including its concentration, the shear-thinning nature of the drop fluid, and the thickness of the double layer, on the electrophoretic behavior of a drop are discussed. In general, the more significant the shear-thinning nature of the drop fluid is, the larger the mobility is, and this effect is pronounced as the thickness of the double layer decreases. However, if the double layer is sufficiently thick, this effect becomes negligible. In general, the higher the concentration of drops is, the smaller the mobility is; however, if the double layer is either sufficiently thin or sufficiently thick, this effect becomes unimportant.

Electrophoresis of a concentrated dispersion of spherical particles covered by an ion-penetrable membrane layer

The electrophoretic behavior of a concentrated dispersion of soft spherical particles is investigated theoretically, taking the effects of double-layer overlapping and double-layer polarization into account. Here, a particle comprises a rigid core and an ion-penetrable layer containing fixed charge, which mimics biocolloids and particles covered by artificial membrane layers. A cell model is adopted to simulate the system under consideration, and a pseudo-spectral method based on Chebyshev polynomials is chosen for the resolution of the governing electrokinetic equations. The influence of the key parameters, including the thickness of the double layer, the concentration of particles, the surface potential of the rigid core of a particle, and the thickness, the amount of fixed charge, and the friction coefficient of the membrane layer of a particle on the electrophoretic behavior of the system under consideration is discussed. We show that while the result for the case of a dispersion containing rigid particles can be recovered as the limiting case of a dispersion containing soft particles, qualitative behaviors that are not present in the former are observed in the latter.

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 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 concentrated dispersion of spherical particles in a non-Newtonian fluid

Electrophoresis is one of the most widely used analytical tools for the quantification of the charged conditions on the surface of fine particles including biological entities. Although it has been studied extensively in the past, relevant results for the case when the dispersion medium is non-Newtonian are very limited. This may occur, for example, when the concentration of the dispersed phase is not low, which is not uncommon in practice. Here, the electrophoresis of a concentrated spherical dispersion in a Carreau fluid is analyzed theoretically under the conditions of low electric potential and weak external applied electrical field. A pseudospectral method coupled with a Newton-Raphson iteration procedure is used to solve the electrokinetic equations describing the phenomenon under consideration. We conclude that the more significant the shear thinning effect of the fluid, the larger the mobility, and this phenomenon is pronounced for the case when the double layer surrounding a particle is thin. We show that if the double layer is thin and the effect of shear thinning is significant, a second vortex can be observed in the neighborhood of a particle.