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.     

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 a Soft Particle

A theory of the dynamic electrophoretic mobility of a spherical soft particle (that is, a polyelectrolyte-coated spherical particle) in an oscillating electric field is presented. In the absence of the polyelectrolyte layer a spherical soft particle becomes a spherical hard particle, while in the absence of the particle core it tends to a spherical polyelectrolyte. The present theory thus covers two extreme cases, that is, dynamic electrophoresis of hard particles and that of spherical polyelectrolytes. Simple analytic mobility expressions are derived. It is shown how the dynamic electrophoretic mobility of a soft particle depends on the volume charge density distributed in the polyelectrolyte layer, on the frictional coefficient characterizing the frictional forces exerted by the polymer segments on the liquid flow in the polyelectrolyte layer, on the particle size, and on the frequency of the applied oscillating electric field. Copyright 2001 Academic Press.     

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     

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.     

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.     

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.     

On the electrophoretic mobility of a cylindrical soft particle

A previous theory for the electrophoresis of a cylindrical soft particle (that is, a cylindrical hard particle covered with a layer of polyelectrolytes) [7], which makes use of the condition that the electrical force acting on the polymer segments is balanced with a frictional force exerted by the liquid flow, is modified by replacing this condition with an alternative and more appropriate boundary condition that pressure is continuous at the boundary between the surface layer and the surrounding electrolyte solution. The general mobility expression thus obtained is found to reproduce all of the approximate analytic mobility expressions derived previously. Received: 20 July 2000/Accepted: 21 August 2000     

Effects of dielectric gradients-mediated ions partitioning on the electrophoresis of composite soft particles: An analytical theory

In this work, we report original analytical expressions defining the electrophoretic mobility of composite soft particles comprising an inner core and a surrounding polymer shell with differentiated permeabilities to ions from aqueous background electrolyte and to fluid flow developed under applied DC field conditions. The existence of dielectric permittivity gradients operational at the core/shell and shell/solution interfaces is accounted for within the Debye–Hückel approximation and flat plate configuration valid in the thin double layer regime. The proposed electrophoretic mobility expressions, applicable to weakly to moderately charged particles with size well exceeding the Debye layer thickness, involve the relevant parameters describing the particle core/shell structure and the electrohydrodynamic features of the core and shell particle components. It is shown that the analytical expressions reported so far in literature for the mobility of hard (impermeable) or porous particles correspond to asymptotic limits of the more generic results detailed here. The impacts of dielectric-mediated effects of ions partitioning between bulk solution and particle body on the electrophoretic response are further discussed. The obtained expressions pave the way for a refined quantitative, analytical interpretation of electrophoretic mobility data collected on soft (nano)particles (e.g., functionalized dendrimers and multilayered polyelectrolytic particles) or biological cells (e.g., viruses) for which the classical hard core-soft shell representation is not appropriate.     

Electrostatic interaction between soft particles

Electrostatic interaction between two soft particles (i.e., polyelectrolyte-coated particles) in an electrolyte solution is discussed. An approximate analytic expression for the interaction energy between two dissimilar soft spheres is derived by applying Derjaguin's approximation to the corresponding interaction energy between two parallel dissimilar soft plates for the case where the density of fixed charges within the polyelectrolyte layer is low. The obtained expression covers various limiting cases that include hard sphere/hard sphere interaction, spherical polyelectrolyte/spherical polyelectrolyte interaction, soft sphere/spherical polyelectrolyte interaction, soft sphere/hard sphere interaction, and spherical polyelectrolyte/hard sphere interaction.     

Electrokinetics of soft particles

Theories of electrokinetics of soft particles, which are particles covered with an ion-penetrable surface layer of polyelectrolytes, are reviewed. Approximate analytic expressions are given, which describe various electrokinetics of soft particles both in dilute and concentrated suspensions, that is, electrophoretic mobility, electrical conductivity, sedimentation velocity and potential, dynamic electrophoretic mobility, colloid vibration potential, and electrophoretic mobility under salt-free condition.     

Hard versus soft particle electrokinetics of silica colloids

To verify the existence of a gel layer at the surface of silica, dependences of the electrophoretic mobility of fresh and aged colloidal silica particles on the KCl concentration are measured. These dependences, corrected for the relaxation/polarization effect, are fitted by analytical expressions based on the model of hard, soft, and brush surfaces. A bad fit is obtained for both silicas when its surface is considered ideal (hard). Much better fits are achieved with the invariable soft layer model for the fresh silica but especially for the aged silica whose surface is less charged probably as a result of an extension and/or loosening of the layer. A perfect fit is found for aged silica when applying a trivial model of the soft polyelectrolyte layer combined with the scaling model of polyelectrolyte brushes.     

Analysis of the response of suspended colloidal soft particles to a constant electric field

A network model, originally designed for an electrokinetic study of soft particle suspensions, has been used for an in-depth analysis of the physical behavior of these systems under the action of an externally applied DC electric field. The versatility of the network simulation method used makes it possible to obtain information readily not only about the electrophoretic mobility, but also about any physical variable of interest at all points around the suspended particle: electric potential, ion concentrations, fluid velocity. The field-induced polarization of the double layer is described in terms of the dependence of these and other derived variables (volume charge density, electric field components, ion flux components) on the distance to the membrane-solution interface. In contrast to colloidal suspensions of hard particles, which basically depend on just two parameters (the reciprocal Debye length multiplied by the particle radius, kappaa, and the zeta potential, zeta), soft particle suspensions require a wider parameter set. First, there are two characteristic diffusion lengths in the system (one inside the membrane and the other in the solution) and two geometrical lengths (the core radius a and the membrane thickness (b-a)). Furthermore, there is the fixed charge density inside the membrane (and possibly a surface charge density over the core) that cannot be represented by a zeta potential. Finally, the parameter that characterizes the interaction between the fluid and the permeable membrane, gamma, strongly influences the behavior of the system. Dependences on all these parameters (except the geometrical ones) are included in this study.     

Electrophoretic mobility of colloidal particles coated with a layer of adsorbed polymers

A general expression is given for the electrophoretic mobility of a large charged colloidal particle coated with a layer of adsorbed charged polymers. A liquid flow within the polymer layer is taken into account. The potential distribution is calculated on the basis of the non-linear Poisson Boltzmann equation. Simple approximate analytic expressions for the electrophoretic mobility are derived for various cases.     

Primary electroviscous effect in a dilute suspension of soft particles

A theory for the primary electroviscous effect in a dilute suspension of soft particles (i.e., particles coated with an ion-penetrable surface layer of polyelectrolytes) in an electrolyte solution is presented. The general expression for the effective viscosity eta s of the suspension and the primary electroviscous coefficient p, which is further expressed in terms of a function L, is given. On the basis of the general expressions, we derive approximate analytic expressions for eta s and p, which are applicable when the density of the fixed charges distributed within the surface layer is low. Further we obtain a simple approximate analytic expression (without involving numerical integrations) for p applicable for most practical cases. It is found that the function L exhibits a minimum when plotted as a function of kappa a (kappa is the Debye-Hückel parameter and a is the particle core radius), unlike the case of a suspension of hard particles, in which case L decreases as kappa a increases, exhibiting no minimum. The presence of a minimum for the case of a suspension of soft particles is due to the fact that L is proportional to 1/kappa 2 at small kappa a and to kappa 2 at large kappa a. Because of the presence of this minimum, the difference in L between soft and hard particles becomes very large for large kappa a.     

Numerical calculation of the electrophoretic mobility of concentrated suspensions of soft particles

The electrophoretic mobility of spherical soft particles in concentrated colloidal suspensions is numerically calculated. The particle is modeled as a hard core coated with an ion-penetrable membrane bearing a uniform distribution of fixed charges, while the high particle concentration is taken into account by means of a cell model. The network simulation method used makes it possible to solve the problem without any restrictions on the values of the parameters such as particle concentration, membrane thickness, fixed charge density in the membrane, viscous drag in the membrane, number and valence of ionic species, electrolyte concentration, etc. The theoretical model used is similar to the one presented by Ohshima [H. Ohshima, J. Colloid Interface Sci. 225 (2000) 233], except for the use of the Shilov-Zharkikh, rather than the Levine-Neale, boundary condition for the electric potential, and the inclusion in the force balance equation of an additional term corresponding to the force exerted by the liquid on the core of the moving particle [J.J. López-García, C. Grosse, J. Horno, J. Colloid Interface Sci. 265 (2003) 327]. The obtained results only coincide with existing analytical expressions for low particle concentrations, low particle charge, and when the electrolyte concentration is high, the membrane is thick, and its resistance to the fluid flow is high. This suggests that most interpretations of the electrophoretic mobility of soft particles in concentrated suspensions require numerical calculations.     

Electrophoretic mobility of a soft particle in a salt-free medium

A theory is presented for the electrophoretic mobility mu of dilute spherical soft particles (i.e., polyelectrolyte-coated particles) in salt-free media containing only counterions. As in the case of other types of particles (rigid particles and liquid drops) in salt-free media, there is a certain critical value of the particle charge separating two cases, the low-surface-charge case and the high-surface-charge case. For the low-charge case, the mobility is proportional to the particle charge and coincides with that of a soft particle in an electrolyte solution in the limit of very low electrolyte concentrations kappa-->0 (Hückel's limit), where kappa is the Debye-Hückel parameter. For the high-charge case, however, mu becomes essentially constant, independent of the particle charge, due to the counterion condensation effect.     

Electrophoresis of soft particles with hydrophobic inner core grafted with pH-regulated and highly charged polyelectrolyte layer

Electrophoresis of core–shell composite soft particles possessing hydrophobic inner core grafted with highly charged polyelectrolyte layer (PEL) has been studied analytically. The PEL bears pH-dependent charge properties due to the presence of zwitterionic functional groups. The dielectric permittivity of the PEL and bulk aqueous medium were taken to be different, which resulted in the ion-partitioning effect. Objective of this study was to provide a simple expression for the mobility of such core–shell soft particles under Donnan limit where the thickness of the PEL well exceeds the electric double layer thickness. Going beyond the widely used Debye–Hückel linearization, the nonlinear Poisson–Boltzmann equation coupled with Stokes–Darcy–Brinkman equations was solved to determine the electrophoretic mobility. The derived expression further recovers all the existing results for the electrophoretic mobility under various simplified cases. The graphical presentation of the results illustrated the impact of pertinent parameters on the electrophoretic mobility of such a soft particle.     

Transient electrophoresis of dielectric spheres

The dynamic electrophoretic response of a spherical dielectric particle suspended in an electrolyte solution to a step change in the applied electrics field is analytically studied. The electrical double layer surrounding the particle may have either a small but finite thickness or a very large thickness relative to the particle radius. For the case of electrophoresis of a particle with a thin double layer, the local electroosmotic velocity at the outer edge of the double layer evolving with time after the external field is imposed is used as an apparent slip boundary condition at the particle surface so that the unsteady equation of motion for the fluid flow outside the double layer is solved. Closed-form formulas for the transient electrophoretic mobility of the particle are derived as functions of relevant parameters. The results demonstrate that, when the double layer surrounding the particle is relatively thin, the normalized electrophoretic mobility at a given dimensionless time decreases monotonically with a decrease in the parameter kappaa, where kappa(-1) is the Debye screening length and a is the particle radius. When the double layer of the particle is relatively thick, the particle mobility can have magnitudes comparable to those for a particle with a thin double layer in the initial stage, but will become much smaller afterward. In general, the effect of the relaxation time for transient electrophoresis is negligible, regardless of the value of kappaa.     

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.