Sedimentation of a Charged Soft Sphere within a Charged Spherical Cavity.

The sedimentation of a soft particle composed of an uncharged hard sphere core and a charged porous surface layer inside a concentric charged spherical cavity full of a symmetric electrolyte solution is analyzed in a quasi-steady state. By using a regular perturbation method with small fixed charge densities of the soft sphere and cavity wall, a set of linearized electrokinetic equations relevant to the fluid velocity field, electrical potential profile, and ionic electrochemical potential energy distributions are solved. A closed-form formula for the sedimentation velocity of the soft sphere is obtained as a function of the ratios of core-to-particle radii, particle-to-cavity radii, particle radius-to-Debye screening length, and particle radius-to-porous layer permeation length. The existence of the surface charge on the cavity wall increases the settling velocity of the charged soft sphere, principally because of the electroosmotic enhancement of fluid recirculation within the cavity induced by the sedimentation potential gradient. When the porous layer space charge and cavity wall surface charge have the same sign, the particle velocity is generally enhanced by the presence of the cavity. When these fixed charges have opposite signs, the particle velocity will be enhanced/reduced by the presence of the cavity if the wall surface charge density is sufficiently large/small relative to the porous layer space charge density in magnitude. The effect of the wall surface charge on the sedimentation of the soft sphere increases with decreases in the ratios of core-to-particle radii, particle-to-cavity radii, and particle radius-to-porous layer permeation length but is not a monotonic function of the ratio of particle radius-to-Debye length.

Electrophoresis and electric conduction in a salt-free suspension of charged particles.

The electrophoresis and electric conduction of a suspension of charged spherical particles in a salt-free solution are analyzed by using a unit cell model. The linearized Poisson-Boltzmann equation (valid for the cases of relatively low surface charge density or high volume fraction of the particles) and Laplace equation are solved for the equilibrium electric potential profile and its perturbation caused by the imposed electric field, respectively, in the fluid containing the counterions only around the particle, and the ionic continuity equation and modified Stokes equations are solved for the electrochemical potential energy and fluid flow fields, respectively. Explicit analytical formulas for the electrophoretic mobility of the particles and effective electric conductivity of the suspension are obtained, and the particle interaction effects on these transport properties are significant and interesting. The scaled zeta potential, electrophoretic mobility, and effective electric conductivity increase monotonically with an increase in the scaled surface charge density of the particles and in general decrease with an increase in the particle volume fraction, keeping each other parameter unchanged. Under the Debye-Hückel approximation, the dependence of the electrophoretic mobility normalized with the surface charge density on the ratio of the particle radius to the Debye screening length and particle volume fraction in a salt-free suspension is same as that in a salt-containing suspension, but the variation of the effective electric conductivity with the particle volume fraction in a salt-free suspension is found to be quite different from that in a suspension containing added electrolyte.

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.

Transient electrophoresis of a charged porous particle.

The starting electrophoretic motion of a porous, uniformly charged, spherical particle, which models a solvent-permeable and ion-penetrable polyelectrolyte coil or floc of nanoparticles, in an arbitrary electrolyte solution due to the sudden application of an electric field is studied for the first time. The unsteady Stokes/Brinkman equations with the electric force term governing the fluid velocity fields are solved by means of the Laplace transform. An analytical formula for the electrophoretic mobility of the porous sphere is obtained as a function of the dimensionless parameters κa , λa , ρp/ρ , and νt/a2 , where a is the radius of the particle, κ is the Debye screening parameter, λ is the reciprocal of the square root of the fluid permeability in the particle, ρp and ρ are the mass densities of the particle and fluid, respectively, ν is the kinematic viscosity of the fluid, and t is the time. The electrophoretic mobility normalized by its steady-state value increases monotonically with increases in νt/a2 and κa , but decreases monotonically with an increase in ρp/ρ , keeping the other parameters unchanged. In general, a porous particle with a high fluid permeability trails behind an identical porous particle with a lower permeability and a corresponding hard particle in the growth of the normalized electrophoretic mobility The normalized electrophoretic acceleration of the porous sphere decreases monotonically with an increase in the time and increases with an increase in λa from zero at λa=0 .

Electrokinetic flow and electric conduction of salt-free solutions in a capillary.

The electrokinetic flow and accompanied electric conduction of a salt-free solution in the axial direction of a charged circular capillary are analyzed. No assumptions are made about the surface charge density (or surface potential) and electrokinetic radius of the capillary, which are interrelated. The Poisson-Boltzmann equation and modified Navier-Stokes equation are solved for the electrostatic potential distribution and fluid velocity profile, respectively. Closed-form formulas for the electroosmotic mobility and electric conductivity in the capillary are derived in terms of the surface charge density. The relative surface potential, electroosmotic mobility, and electric conductivity are monotonic increasing functions of the surface charge density and electrokinetic radius. However, the rises of the relative surface potential and electroosmotic mobility with an increase in the surface charge density are suppressed substantially when it is high due to the effect of counterion condensation. The analytical prediction that the electroosmotic mobility grows with increases in the surface charge density and electrokinetic radius agrees with the experimental results for salt-free solutions in circular microchannels in the literature.

Diffusiophoresis of a charged particle in a microtube.

The diffusiophoresis of a charged sphere along the axis of a circular microtube filled with an electrolyte solution is studied theoretically. The tube wall may be either nonconductive and impermeable or prescribed with a linear electrolyte concentration distribution. The electric double layers at the solid surfaces are thin, but the diffuse-layer polarization effect over the particle surface is considered. The general solutions to the electrokinetic differential equations are expressed in spherical and cylindrical coordinates, whereas the boundary conditions at the particle surface are satisfied by a collocation technique. The collocation solutions for the diffusiophoretic velocity of the particle, which are in good agreement with the asymptotic formula derived from a reflection method, are obtained for various values of the radius ratio and zeta potential ratio between the particle and the microtube and of other relevant parameters. The contributions from the diffusioosmotic flow along the tube wall and wall-corrected diffusiophoretic driving force to the particle velocity can be superimposed due to the linearity. Although the diffusiophoretic velocity in an uncharged microtube is in general a decreasing function of the particle-to-tube radius ratio and can reverse its direction, it can increase with increases in this ratio due to the competition of the wall effects of possible electrochemical enhancement and hydrodynamic retardation to the particle motion. When the zeta potentials associated with the tube and particle are equivalent, the diffusioosmotic flow induced by the tube wall dominates the diffusiophoretic motion.

Diffusiophoretic mobility of charge-regulating porous particles.

The diffusiophoresis of a charge-regulating porous sphere, such as polyelectrolyte coil, with an arbitrary thickness of the electric double layer in an electrolyte solution prescribed with a concentration gradient is analytically studied for the first time. The ionogenic functional groups and hydrodynamic frictional segments distribute uniformly within the permeable particle, and a charge regulation model for the association and dissociation reactions of the functional groups relates the fixed charge density to the local electric potential. The electrokinetic equations governing the electric potential, ionic electrochemical potential, and fluid velocity distributions are solved as power-series expansions in the basic fixed charge density. An explicit formula for the diffusiophoretic mobility of the particle, which vanishes at the isoelectric point, is derived from a force balance. The effects of charge regulation on the diffusiophoretic mobility, which depend on various particle and electrolyte characteristics such as the reaction equilibrium constants of the ionogenic functional groups, are significant and interesting. The variation in the bulk concentration of the charge-determining ions can produce more than one reversal in the direction of the diffusiophoretic velocity. The obtained results differ conspicuously from those of impermeable particles and provide valuable information for the interpretation of experimental data.

Startup of electrophoresis in a suspension of colloidal spheres.

The transient electrophoretic response of a homogeneous suspension of spherical particles to the step application of an electric field is analyzed. The electric double layer encompassing each particle is assumed to be thin but finite, and the effect of dynamic electroosmosis within it is incorporated. The momentum equation for the fluid outside the double layers is solved through the use of a unit cell model. Closed-form formulas for the time-evolving electrophoretic and settling velocities of the particles in the Laplace transform are obtained in terms of the electrokinetic radius, relative mass density, and volume fraction of the particles. The time scale for the development of electrophoresis and sedimentation is significantly smaller for a suspension with a higher particle volume fraction or a smaller particle-to-fluid density ratio, and the electrophoretic mobility at any instant increases with an increase in the electrokinetic particle radius. The transient electrophoretic mobility is a decreasing function of the particle volume fraction if the particle-to-fluid density ratio is relatively small, but it may increase with an increase in the particle volume fraction if this density ratio is relatively large. The particle interaction effect in a suspension on the transient electrophoresis is much weaker than that on the transient sedimentation of the particles.

Start-up of electrophoresis of an arbitrarily oriented dielectric cylinder.

An analytical study is presented for the transient electrophoretic response of a circular cylindrical particle to the step application of an electric field. The electric double layer adjacent to the particle surface is thin but finite compared with the radius of the particle. The time-evolving electroosmotic velocity at the outer boundary of the double layer is utilized as a slip condition so that the transient momentum conservation equation for the bulk fluid flow is solved. Explicit formulas for the unsteady electrophoretic velocity of the particle are obtained for both axially and transversely applied electric fields, and can be linearly superimposed for an arbitrarily-oriented applied field. If the cylindrical particle is neutrally buoyant in the suspending fluid, the transient electrophoretic velocity is independent of the orientation of the particle relative to the applied electric field and will be in the direction of the applied field. If the particle is different in density from the fluid, then the direction of electrophoresis will not coincide with that of the applied field until the steady state is attained. The growth of the electrophoretic mobility with the elapsed time for a cylindrical particle is substantially slower than for a spherical particle.

Magnetohydrodynamic motion of a colloidal sphere with self-electrochemical surface reactions in a spherical cavity.

An analytical study is presented for the magnetic-field-induced motion of a colloidal sphere with spontaneous electrochemical reactions on its surface situated at the center of a spherical cavity filled with an electrolyte solution at the quasi-steady state. The zeta potential associated with the particle surface may have an arbitrary distribution, whereas the electric double layers adjoining the particle and cavity surfaces are taken to be thin relative to the particle size and the spacing between the solid surfaces. The electric current and magnetic flux density distributions are solved for the particle and fluid phases of arbitrary electric conductivities and magnetic permeabilities. Applying a generalized reciprocal theorem to the Stokes equations with a Lorentz force term resulting from these density distributions for the fluid motion, we obtain explicit formulas for the translational and angular velocities of the colloidal sphere valid for all values of the particle-to-cavity size ratio. The particle velocities decrease monotonically with an increase in this size ratio. For the limiting case of an infinitely large cavity, our result reduces to the relevant solution for an unconfined spherical particle. The boundary effect on the movement of the particle with interfacial self-electrochemical reactions induced by the magnetohydrodynamic force is equivalent to that in sedimentation and much stronger than that in general phoretic motions.

Motion of a colloidal sphere with interfacial self-electrochemical reactions induced by a magnetic field.

The motion of a spherical colloidal particle with spontaneous electrochemical reactions occurring on its surface in an ionic solution subjected to an applied magnetic field is analyzed for an arbitrary zeta potential distribution. The thickness of the electric double layer adjacent to the particle surface is assumed to be much less than the particle radius. The solutions of the Laplace equations governing the magnetic scalar potential and electric potential, respectively, lead to the magnetic flux and electric current density distributions in the particle and fluid phases of arbitrary magnetic permeabilities and electric conductivities. The Stokes equations modified with the Lorentz force contribution for the fluid motion are dealt by using a generalized reciprocal theorem, and closed-form formulas for the translational and angular velocities of the colloidal sphere induced by the magnetohydrodynamic effect are obtained. The dipole and quadrupole moments of the zeta potential distribution over the particle surface cause the particle translation and rotation, respectively. The induced velocities of the particle are unexpectedly significant, and their dependence on the characteristics of the particle-fluid system is physically different from that for electromagnetophoretic particles or phoretic swimmers.

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.

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.

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.

Start-Up Electrophoresis of a Cylindrical Particle with Arbitrary Double Layer Thickness.

The start-up of electrophoretic motion of a charged circular cylindrical particle in an unbounded solution of arbitrary electrolytes is analytically investigated. The modified Stokes equation for the transient fluid flow field is solved by using the Laplace transform. Analytical formulas for the time-evolving electrophoretic velocities of the dielectric cylinder are determined for the transversely and axially imposed electric fields, and they can be superimposed linearly for an imposed electric field of arbitrary direction. The transient electrophoretic velocities normalized by their respective steady-state values increase monotonically with an increase in the ratio of the particle radius to the Debye screening length but decrease monotonically with an increase in the particle-to-fluid density ratio, keeping the other parameter unchanged. The normalized electrophoretic acceleration of the particle decreases monotonically with the elapsed time. In general, the electrophoretic velocity of the cylindrical particle is not collinear with the arbitrarily oriented imposed electric field. The effect of the relaxation time for the transient electrophoresis is much more important for a cylindrical particle than for a spherical particle.

Sedimentation Velocity and Potential in Dilute Suspensions of Charge-Regulating Porous Spheres.

The sedimentation of a charge-regulating porous sphere surrounded by an arbitrary electric double layer, which usually models a permeable polyelectrolyte coil or an aggregate of nanoparticles, is analyzed for the first time. The hydrodynamic frictional segments and ionogenic functional groups uniformly distribute in the porous sphere, and a regulation mechanism for the dissociation and association reactions occurring at these functional groups linearly relates the local electric potential to fixed charge density. The linearized electrokinetic equations governing the ionic concentration (or electrochemical potential energy), electric potential, and fluid velocity fields are solved for the case of a small basic fixed charge density by the regular perturbation method. Analytical formulae for the sedimentation velocity of a porous sphere and sedimentation potential of a dilute suspension of porous spheres are then obtained. The charge regulation tends to reduce the electrokinetic retardation to sedimentation velocity and the sedimentation potential (can be as much as 50 and 25%, respectively) compared to the case that the fixed charge density is a constant. Both the electrokinetic retardation to sedimentation velocity and the sedimentation potential vanish at the isoelectric point of the particles. The increase in the bulk concentration of the potential-determining ions crossing the isoelectric point changes signs of the fixed charges and thus causes a reversal in the direction of the sedimentation potential. The effects of charge regulation on the sedimentation of porous particles differ substantially from those of hard particles.

Electrokinetic Flow of Salt-Free Solutions in a Fibrous Porous Medium.

The electrokinetic flow of a salt-free solution in the fibrous porous medium constituted by an array of parallel charged circular cylinders subject to a pressure gradient and an electric field imposed in the axial direction is analytically studied via the use of a unit cell model. The Poisson-Boltzmann equation and modified Navier-Stokes equation applicable to a unit cell accommodating the salt-free solution around an individual cylinder are solved to determine the electric potential profile and fluid velocity distribution. Results of the electroosmotic velocity and effective electric conductivity in the fiber matrix are obtained as functions of the surface charge density of the dielectric cylinders and the porosity of the fiber matrix. The effects of the porosity or interactions among the cylinders on the electric potential distribution, electroosmotic velocity, and effective electric conductivity are significant and interesting under practical conditions. The apparent zeta potential, electroosmotic velocity, and effective electric conductivity increase monotonically with an increase in the surface charge density of the cylinders. When the porosity of the fiber matrix and surface charge density of the cylinders are high, the increases of the apparent zeta potential and electroosmotic velocity with the surface charge density are substantially suppressed due to the counterion condensation effect. However, this effect becomes weak when the porosity is low.

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.

Diffusiophoresis of a Charged Porous Particle in a Charged Cavity.

The quasi-steady diffusiophoresis of a charged porous sphere situated at the center of a charged spherical cavity filled with a liquid solution of a symmetric electrolyte is analyzed. The porous particle can represent a solvent-permeable and ion-penetrable polyelectrolyte molecule or floc of nanoparticles in which fixed charges and frictional segments are uniformly distributed, whereas the spherical cavity can denote a charged pore involved in microfluidic or drug-delivery systems. The linearized electrokinetic differential equations governing the ionic concentration, electric potential, and fluid velocity distributions in the system are solved by using a perturbation method with the fixed charge density of the particle and the ζ-potential of the cavity wall as the small perturbation parameters. An expression for the diffusiophoretic (electrophoretic and chemiphoretic) mobility of the confined particle with arbitrary values of a/ b, κ a, and λ a is obtained in closed form, where a and b are the radii of the particle and cavity, respectively; κ and λ are the reciprocals of the Debye screening length and the length characterizing the extent of flow penetration into the porous particle, respectively. The presence of the charged cavity wall significantly affects the diffusiophoretic motion of the particle in typical cases. The diffusio-osmotic (electro-osmotic and chemiosmotic) flow occurring at the cavity wall can substantially alter the particle velocity and even reverse the direction of diffusiophoresis. In general, the particle velocity decreases with an increase in a/ b, increases with an increase in κ a, and decreases with an increase in λ a, but exceptions exist.

Sedimentation Velocity and Potential in Dilute Suspensions of Charged Porous Shells.

The sedimentation of a charged spherical porous shell with arbitrary inner and outer radii, which can model a permeable microcapsule or vesicle, in a general electrolyte solution is analytically examined. The relaxation effect in the electric double layers of arbitrary thickness around the porous shell is considered. The differential equations governing the electric potential profile, ionic electrochemical potential energy (or concentration) distributions, and fluid velocity field are linearized by taking the system to be only slightly distorted from equilibrium. These linearized equations are solved using a perturbation method with the density of the fixed charge of the porous shell as the small perturbation parameter. Closed-form formulas for the sedimentation velocity of a porous shell and sedimentation potential in a suspension of porous shells are obtained from a force balance and a zero current requirement, respectively. Both the charge-induced sedimentation velocity retardation and sedimentation potential are monotonic increasing functions of the relative shell thickness, and these increases are substantial if the shell is thin. The sedimentation velocity and potential are complex functions of the electrokinetic radius and normalized flow penetration length of the porous shell. In the limit of the porous shells with zero inner radius, our formulas for the sedimentation velocity and potential reduce to the results obtained for the intact porous spheres.