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.     

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.     

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 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.     

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 charge-regulated particle at an arbitrary position in a spherical cavity

The boundary effect on the electrophoretic behavior of a particle is examined by considering a sphere at an arbitrary position in a spherical cavity for the case of low electrical potential and weak applied electric field. Here, a charge-regulated model is used to describe the charge conditions on the particle surface. This model finds practical applications where the behavior of biocolloids such as cells or microorganisms and entities covered by an artificial membrane need to be simulated. The two idealized models often used in relevant studies can be recovered as the limiting cases of the present model.     

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

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

Electrophoresis of a colloidal sphere in a spherical cavity with arbitrary zeta potential distributions

An analytical study is presented for the quasi-steady electrophoretic motion of a dielectric sphere situated at the center of a spherical cavity when the surface potentials are arbitrarily nonuniform. The applied electric field is constant, and the electric double layers adjacent to the solid surfaces are assumed to be much thinner than the particle radius and the gap width between the surfaces. The presence of the cavity 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 the viscous retardation of the moving particle; and (3) a circulating electroosmotic flow of the suspending fluid exists because of the interaction between the electric field and the charged wall. The Laplace and Stokes equations are solved analytically for the electric potential and velocity fields, respectively, in the fluid phase, and explicit formulas for the electrophoretic and angular velocities of the particle are obtained. To apply these formulas, one has to calculate only the monopole, dipole, and quadrupole moments of the zeta-potential distributions at the particle and cavity surfaces. It is found that the contribution from the electroosmotic flow developing from the interaction of the imposed electric field with the thin double layer adjacent to the cavity wall and the contribution from the wall-corrected electrophoretic driving force to the particle velocities can be superimposed as a result of the linearity of the problem.     

Electrophoresis of a colloidal sphere in a spherical cavity with arbitrary zeta potential distributions and arbitrary double-layer thickness

The electrophoretic motion of a dielectric sphere situated at the center of a spherical cavity with an arbitrary thickness of the electric double layers adjacent to the particle and cavity surfaces is analyzed at the quasisteady state when the zeta potentials associated with the solid surfaces are arbitrarily nonuniform. Through the use of the multipole expansions of the zeta potentials and the linearized Poisson-Boltzmann equation, the equilibrium double-layer potential distribution and its perturbation caused by the applied electric field are separately solved. The modified Stokes equations governing the fluid velocity field are dealt with using a generalized reciprocal theorem, and explicit formulas for the electrophoretic and angular velocities of the particle valid for all values of the particle-to-cavity size ratio are obtained. To apply these formulas, one only has to calculate the monopole, dipole, and quadrupole moments of the zeta potential distributions at the particle and cavity surfaces. In some limiting cases, our result reduces to the analytical solutions available in the literature. In general, the boundary effect on the electrophoretic motion of the particle is a qualitatively and quantitatively sensible function of the thickness of the electric double layers relative to the radius of the cavity.     

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.     

Electrophoresis in a non-Newtonian fluid: sphere in a spherical cavity

The electrophoretic behavior of a sphere in a non-Newtonian fluid is investigated theoretically by analyzing the phenomenon that occurs in a spherical cavity under the condition of a weak applied electrical field. Non-Newtonian behavior in the liquid phase may be due to, for example, the addition of polymer to a colloidal dispersion to improve its stability. It may also arise from the increase in the volume fraction of the dispersed phase such as the slurry used in chemical mechanical polishing. A Carreau model is adopted to characterize the shear-thinning behavior of the liquid phase. We show that the difference between the mobility of the particle based on the present model and that based on the corresponding Newtonian fluid increases with the decrease in the thickness of a double layer. The shear-thinning nature of the liquid phase has the effect of increasing the mobility.     

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.     

Magnetohydrodynamic effects on a charged colloidal sphere with arbitrary double-layer thickness

An analytical study is presented for the magnetohydrodynamic (MHD) effects on a translating and rotating colloidal sphere in an arbitrary electrolyte solution prescribed with a general flow field and a uniform magnetic field at a steady state. The electric double layer surrounding the charged particle may have an arbitrary thickness relative to the particle radius. Through the use of a simple perturbation method, the Stokes equations modified with an electric force term, including the Lorentz force contribution, are dealt by using a generalized reciprocal theorem. Using the equilibrium double-layer potential distribution from solving the linearized Poisson-Boltzmann equation, we obtain closed-form formulas for the translational and angular velocities of the spherical particle induced by the MHD effects to the leading order. It is found that the MHD effects on the particle movement associated with the translation and rotation of the particle and the ambient fluid are monotonically increasing functions of κa, where κ is the Debye screening parameter and a is the particle radius. Any pure rotational Stokes flow of the electrolyte solution in the presence of the magnetic field exerts no MHD effect on the particle directly in the case of a very thick double layer (κa→0). The MHD effect caused by the pure straining flow of the electrolyte solution can drive the particle to rotate, but it makes no contribution to the translation of the particle.     

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.     

High-order field electrophoresis theory for a nonuniformly charged sphere

An electrophoresis theory is developed for a rigid sphere in a general nonuniform electric field. The zeta potential distribution and the double-layer thickness are both arbitrary. The zeta potential of the sphere is assumed to be small so that the deformation of the double layer can be neglected. Explicit expressions for the translational and rotational velocities of the sphere are derived in terms of the multipole moments of the zeta potential distribution and the tensor coefficients of the applied electric field. The presence of the kth-order component in the electrical potential field applied to the sphere results in a translation of the sphere only when the sphere possesses the (k-1)th- or (k+1)th-order multipole moments of the zeta potential distribution. In addition, the kth-order component in the electrical potential field causes a rotation of the sphere only when the sphere possesses the kth-order moment of the zeta potential distribution. As an illustrative example for the utility of our theory, we theoretically devise an electrophoresis analysis scheme for estimating the dipole moment of a dipolar sphere by observing the electrophoretic translation of the sphere in a quadratic potential field.     

Transient electrophoresis of spherical particles at low potential and arbitrary double-layer thickness

A theoretical study is presented for the dynamic electrophoretic response of a charged spherical particle in an unbounded electrolyte solution to a step change in the applied electric field. The electric double layer surrounding the particle may have an arbitrary thickness relative to the particle radius. The transient Stokes equations modified with the electrostatic effect which govern the fluid velocity field are linearized by assuming that the system is only slightly distorted from equilibrium. Semianalytical results for the transient electrophoretic mobility of the particle are obtained as a function of relevant parameters by using the Debye-Huckel approximation. The results demonstrate that the electrophoretic mobility of a particle with a constant relative mass density at a specified dimensionless time normalized by its steady-state quantity decreases monotonically with a decrease in the parameter kappaa, where kappa(-1) is the Debye screening length and a is the particle radius. For a given value of kappaa, a heavier particle lags behind a lighter one in the development of the electrophoretic mobility. In the limits of kappaa --> infinity and kappaa = 0, our results reduce to the corresponding analytical solutions available in the literature. The electrophoretic acceleration of the particle is a monotonic decreasing function of the time for any fixed value of kappaa. In practical applications, the effect of the relaxation time for the transient electrophoresis is negligible, regardless of the value of kappaa or the relative mass density of the particle.     

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.     

Diffusiophoresis and electrophoresis of a charged sphere perpendicular to two plane walls

The problem of diffusiophoretic and electrophoretic motions of a dielectric spherical particle in an electrolyte solution situated at an arbitrary position between two infinite parallel plane walls is studied theoretically in the quasisteady limit of negligible Peclet and Reynolds numbers. The applied electrolyte concentration gradient or electric field is uniform and perpendicular to the plane walls. The electric double layer at the particle surface is assumed to be thin relative to the particle radius and to the particle-wall gap widths, but the polarization effect of the diffuse ions in the double layer is incorporated. To solve the conservative equations, the general solution is constructed from the fundamental solutions in both cylindrical and spherical coordinates. The boundary conditions are enforced first at the plane walls by the Hankel transforms and then on the particle surface by a collocation technique. Numerical results for the diffusiophoretic and electrophoretic velocities of the particle relative to those of a particle under identical conditions in an unbounded solution are presented for various cases. The collocation results agree well with the approximate analytical solutions obtained by using a method of reflections. The presence of the walls can reduce or enhance the particle velocity, depending on the properties of the particle-solution system and the relative particle-wall separation distances. The boundary effects on diffusiophoresis and electrophoresis of a particle normal to two plane walls are found to be quite significant and complicated, and generally stronger than those parallel to the confining walls.     

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.