Electrophoretic mobility of cylindrical soft particles

 A general theory for the electrophoresis of a cylindrical soft particle (i.e., a cylindrical hard colloidal particle coated with a layer of ion-penetrable polyelectrolytes) in an electrolyte solution in an applied transverse or tangential electric field is proposed. This theory unites two different electrophoresis theories for cylindrical hard particles and for cylindrical polyelectrolytes. That is, the general mobility expression obtained in this paper tends to the mobility expression for a cylindrical hard particle for the case where the polyelectrolyte layer is absent or the frictional coefficient in the poly-electrolyte layer becomes infinity, whereas it tends to that for a cylin-drical polyelectrolyte in the absence of the particle core. Simple approximate analytic mobility expressions are also presented. Received: 29 August 1996 Accepted: 7 November 1996     

On the General Expression for the Electrophoretic Mobility of a Soft Particle

Electrokinetic equations for electrophoresis of a soft particle (that is, a hard particle covered with a layer of polyelectrolytes) have been solved previously under the conditions that the net force acting on the soft particle as a whole (the particle core plus the polyelectrolyte layer) must be zero and that the electrical force acting on the polymer segment is balanced with a frictional force exerted by the liquid flow (J. Colloid Interface Sci. 163, 474 (1994)). In the present work we replaced the latter condition by the alternative and more appropriate condition that pressure is continuous at the boundary between the surface layer and the surrounding electrolyte solution to solve the electrokinetic equations and obtained the general mobility expression for the electrophoretic mobility of a spherical soft particle. It is found that the general mobility expression thus obtained reproduces all of the approximate mobility expressions derived previously and, in addition, that the continuous pressure condition leads to the correct limiting behavior of the electrophoretic mobility in the case where the frictional coefficient tends to zero (this behavior cannot be derived from the force balance condition for the polyelectrolyte layer). Copyright 2000 Academic Press.     

Electrophoresis of soft particles: analytic approximations

An approximate analytic expression is derived for the electrophoretic mobility of a weakly charged spherical soft particle (i.e., a hard particle covered with a weakly charged polyelectrolyte layer) on the basis of the general mobility expression for soft particles (Ohshima, H., J. Colloid Interface Sci. 2000, 228, 190-193). The obtained mobility expression, which reproduces various approximate results so far derived and gives some new mobility formulas, covers all types of weakly charged soft particles with arbitrary values of the thickness of polymer layer, the radius of the particle core, the electrophoretic softness, and the Debye length, including spherical polyelectrolytes with no particle core as well as spherical hard particles with no polyelectrolyte layer.     

Dynamic electrophoretic mobility of spherical colloidal particles in a salt-free medium

A theory is proposed for the dynamic electrophoretic mobility mu(omega) of spherical colloidal particles in a salt-free medium containing only counterions in an oscillating electric field of frequency omega. The dynamic mobility depends on the frequency omega of the applied electric field and on the particle volume fraction as well as on the particle surface charge. It is found that as in the case of the static electrophoretic mobility mu(0) in salt-free media, there is a certain critical value of the particle surface charge separating two cases, that is, the low-surface-charge case and the high-surface-charge case (in the latter case the counterion condensation takes place near the particle surface). For the low-surface-charge case, the dynamic mobility agrees with that of a sphere in an electrolyte solution in the limit of very low electrolyte concentrations kappaa-->0 (Hückel's limit), where kappa is the Debye-Hückel parameter and a is the particle radius. For the high-surface-charge case, however, the dynamic mobility becomes constant independent of the particle surface charge, because of the counterion condensation effects. A simple expression for the ratio mu(omega)/mu(0) applicable for all cases is given.     

Dynamic electrophoretic mobility of spherical colloidal particles in salt-free concentrated suspensions

In this contribution, the dynamic electrophoretic mobility of spherical colloidal particles in a salt-free concentrated suspension subjected to an oscillating electric field is studied theoretically using a cell model approach. Previous calculations focusing the analysis on cases of very low or very high particle surface charge are analyzed and extended to arbitrary conditions regarding particle surface charge, particle radius, volume fraction, counterion properties, and frequency of the applied electric field (sub-GHz range). Because no limit is imposed on the volume fractions of solids considered, the overlap of double layers of adjacent particles is accounted for. Our results display not only the so-called counterion condensation effect for high particle charge, previously described in the literature, but also its relative influence on the dynamic electrophoretic mobility throughout the whole frequency spectrum. Furthermore, we observe a competition between different relaxation processes related to the complex electric dipole moment induced on the particles by the field, as well as the influence of particle inertia at the high-frequency range. In addition, the influences of volume fraction, particle charge, particle radius, and ionic drag coefficient on the dynamic electrophoretic mobility as a function of frequency are extensively analyzed.     

Dynamic Mobility of Two Spherical Particles with Thick Double Layers

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

Electrical phenomena of soft particles. A soft step function model

Simple analytic expressions are derived for the electrophoretic mobility of a soft particle consisting of the hard particle core covered with an ion-penetrable surface layer of polyelectrolyte for the case where the electric potential is low. The effect of the distribution of the polymer segments is taken into account by modeling the surface layer as a soft step function with the inhomogeneous distribution width δ. It is shown that the electrophoretic mobility becomes lower than that for the hard step function model and that the maximum deviation of the soft step function model from the hard step function model, which is a function of λδ (where 1/λ is the softness parameter) and κ/λ (where κ is the Debye-Hückel parameter), is 2.7% at λδ = 0.1, 5.1% at λδ = 0.2, and 11% at λδ = 0.5. In the limit of very high electrolyte concentrations, the obtained mobility expression tends to the result derived from the conventional hard step function model. In addition, an analytic expression for the interaction energy between two similar soft plates is derived on the basis of the present soft step function model. The magnitude of the interaction energy is shown to decrease by a factor 1/(1 + κδ)(2). Approximate analytic expressions for the interaction energies between two similar soft spheres and between two similar soft cylinders are also derived with the help of Derjaguin's approximation.     

Ion size effects on the electrokinetics of salt-free concentrated suspensions in ac fields

We analyze the influence of finite ion size effects in the response of a salt-free concentrated suspension of spherical particles to an oscillating electric field. Salt-free suspensions are just composed of charged colloidal particles and the added counterions released by the particles to the solution that counterbalance their surface charge. In the frequency domain, we study the dynamic electrophoretic mobility of the particles and the dielectric response of the suspension. We find that the Maxwell-Wagner-O’Konski process associated with the counterions condensation layer is enhanced for moderate to high particle charges, yielding an increment of the mobility for such frequencies. We also find that the increment of the mobility grows with ion size and particle charge. All these facts show the importance of including ion size effects in any extension attempting to improve standard electrokinetic models.     

Electrophoretic Mobility of Soft Particles in Concentrated Suspensions

A general theory is developed for the electrophoretic mobility of spherical soft particles (i.e., spherical hard colloidal particles of radius a coated with a layer of polyelectrolytes of thickness d) in concentrated suspensions in an electrolyte solution as a function of the particle volume fraction φ on the basis of Kuwabara's cell model. In the limit d-->0, the mobility expression obtained tends to that for spherical hard particles in concentrated suspensions, whereas in the limit a-->0, it becomes that for spherical polyelectrolytes (charged porous spheres with no particle core). Simple approximate analytic mobility expressions are derived for the case where relaxation effect is negligible. It is found that in practical cases, the φ dependence of the mobility is negligible for da, the mobility strongly decreases with increasing φ. Copyright 2000 Academic Press.     

Approximate analytic expression for the dynamic electrophoretic mobility of a spherical colloidal particle in an oscillating electric field

An approximate analytic expression is derived for the dynamic electrophoretic mobility of a spherical charged colloidal particle in an electrolyte solution in an applied oscillating electric field. This expression, which takes into account the relaxation effects, is applicable for all values of zeta potential at large kappa a (kappa a > or = ca. 30) and omega/2pi < or = ca. 10 MHz, where kappa is the Debye-Hückel parameter, a is the particle radius, and omega is the frequency of the electric field. It is shown that the obtained mobility expression is in excellent agreement with the exact numerical results of Mangelsdorf and White (J. Chem. Soc., Faraday Trans. 1992, 88, 3567).     

Approximate analytic expressions for the electrophoretic mobility of spherical soft particles

Approximate analytic expressions are derived for the electrophoretic mobility of a weakly charged spherical soft particle consisting of the particle core covered with a surface layer of polymers in an electrolyte solution. The particle core and the surface polymer layer may be charged or uncharged. The obtained electrophoretic mobility expressions, which involve neither numerical integration nor exponential integrals, are found to be in excellent agreement with the exact numerical results. It is also found that the obtained mobility expressions reproduce all the previously derived limiting expressions and approximate analytic expressions for the electrophoretic mobility of a weakly charged spherical soft particle.     

Determination of the dynamic electrophoretic mobility of a spherical colloidal particle through a novel numerical solution of the electrokinetic equations

The standard equations developed to describe the electrophoretic motion of a charged particle immersed in an electrolyte subjected to an oscillating electric field are solved numerically with a new technique suitable for stiff systems. The focus of this work is to use this solution to determine the dynamic particle mobility, one of several quantities that can be extracted from these equations. This solution is valid from low frequencies to indefinitely high frequencies and has no restriction on zeta potential, double-layer thickness, or electrolyte composition. The solution has been used to calculate the dynamic electrophoretic mobility of a particle for a wide range of double-layer thicknesses and zeta potentials. The solution agrees with analytic approximations obtained previously by other authors under the conditions of a thin double layer and low zeta potential. The results are also consistent with calculations valid at frequencies where the ion diffusion length extends a significant distance beyond the double layer as obtained by another numerical technique.     

Dynamic electrophoresis of a spherical dispersion of soft particles subject to a stress-jump condition

The dynamic mobility of a spherical dispersion of soft particles, where a particle comprises a rigid core and a membrane layer, is evaluated for the case when the shear stress across the membrane layer-liquid interface is discontinuous, the so-called stress-jump condition. We show that, due to the effect of double-layer deformation, the magnitude of the dynamic mobility of a particle has a local maximum and the corresponding phase angle has a negative (phase lead) local minimum at a low to medium level of the frequency of the applied electric field. This effect becomes insignificant if the frequency of the applied electric field is sufficiently high. The stress-jump condition may lead to a significant influence on the drag, and consequently the mobility of a particle. The degree of influence depends upon the sign of the stress-jump coefficient and the charged conditions of the membrane layer of the particle.     

Electrophoretic mobility of a charged spherical colloidal particle covered with an uncharged polymer layer

A general expression is derived for the electrophoretic mobility of a spherical charged colloidal particle covered with an uncharged polymer layer in an electrolyte solution in an applied electric field for the case where the particle zeta potential is low. It is assumed that electrolyte ions as well as water molecules can penetrate the polymer layer. Approximate analytic expressions for the electrophoretic mobility of particles carrying low zeta potentials are derived for the two extreme cases in which the particle radius is very large or very small.     

Electrophoresis of soft particles with charged rigid core coated with pH-regulated polyelectrolyte layer

We present a theoretical study on the electrophoresis of a soft particle with a dielectric charged rigid core grafted with a charge-regulated polyelectrolyte layer. The polyelectrolyte layer possesses either an acidic or a basic functional group and the charge dissociation depends on the local pH and ionic concentration of the electrolyte. The dielectric rigid core is considered to possess a uniform volumetric charge density. The electric potential distribution is determined by computing the Poisson-Boltzmann equation outside the core coupled with a Poisson equation inside the impermeable core along with suitable matching conditions at the core-shell interface. The computed electric field is used to determine the mobility of the particle through an existing analytic expression based on the Debye-Huckel approximation. Our results are found to be in good agreement with the existing solutions for the limiting cases. The influence of the core charge density, ionic concentration, and pH of the electrolyte on the particle mobility is studied for different choice of hydrodynamic penetration length of the polyelectrolyte and dissociation constant of the functional group. The critical value of the pH required to achieve zero mobility is estimated. We find that in a monovalent electrolyte solution, the soft particle with a net negative (positive) charge can have positive (negative) mobility.     

Electrohydrodynamics of spherical polyampholyte‐grafted nanoparticles: Multiscale simulations by coupling of molecular dynamics and lattice‐boltzmann method

We perform multiscale simulations based on the coupling of molecular dynamics and lattice‐Boltzmann (LB) method to study the electrohydrodynamics of a polyampholyte‐grafted spherical nanoparticle. The long‐range hydrodynamic interactions are modeled by coupling the movement of particles to a LB fluid. Our results indicate that the net‐neutral soft particle moves with a nonzero mobility under applied electric fields. We systematically explore the effects of different parameters, including the chain length, grafting density, electric field, and charge sequence, on the structures of the polymer layer and the electrophoretic mobility of the soft particle. It shows that the mobility of nanoparticles has remarkable dependence on these parameters. We find that the deformation of the polyampholyte chains and the ion distribution play dominant roles in modulating the electrokinetic behavior of the polyampholyte‐grafted particle. The enhancement or reduction in the accumulation of counterions around monomers can be attributed to the polymer layer structure and the conformational transition of the chains in the electric field. © 2017 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2017 , 55, 1435–1447     

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.     

Colloid vibration potential and ion vibration potential in a dilute suspension of spherical colloidal particles

A general electroacoustic theory is presented for the macroscopic electric field in a dilute suspension of spherical colloidal particles in an electrolyte solution, which consists of the colloid vibration potential (CVP) and the ion vibration potential (IVP), induced by an oscillating pressure gradient field due to an applied sound wave. This is a unified theory that unites previous theories for CVP and those for IVP. Approximate analytic expressions are derived for CVP and IVP. The obtained IVP expression agrees with Debye's formula that is corrected by taking into account the force acting on the electrolyte ions as a result of the pressure gradient in the sound wave. The obtained CVP expression is correct to the first order of the particle zeta potential and applicable for arbitrary kappaalpha, where kappa is the Debye-Hückel parameter and alpha is the particle radius. It is found that an Onsager relation holds between CVP and dynamic electrophoretic mobility. It is also shown that the CVP from particles with very small kappaalpha approaches IVP; that is, in the limit of very small kappaalpha a particle behaves like an ion.     

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.     

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.