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.     

Sedimentation of a concentrated dispersion of composite colloidal particles

The sedimentation of a concentrated spherical dispersion of composite particles, where a particle comprises a rigid core and a membrane layer containing fixed charge, is investigated theoretically. The dispersion is simulated by a unit cell model, and a pseudo-spectral method based on Chebyshev polynomials is adopted to solve the problem numerically. The influences of the thickness of double layer, the concentration of particles, the surface potential of the rigid core of a particle, and the amount of fixed charge in the membrane layer on both the sedimentation potential and the sedimentation velocity are discussed. Several interesting results are observed; for example, depending upon the charged conditions on the rigid core and in the membrane layer of a particle, the sedimentation potential might have both a local maximum and a local minimum and the sedimentation velocity can have a local minimum as the thickness of double layer varies. Also, the sedimentation velocity can have a local maximum as the surface potential varies. We show that the sedimentation potential increases with the concentration of particles. The relation between the sedimentation velocity and the concentration of particles, however, depends upon the thickness of double layer.     

Electrophoretic motion of a soft spherical particle in a nanopore

Many biocolloids, biological cells and micro-organisms are soft particles, consisted with a rigid inner core covered by an ion-permeable porous membrane layer. The electrophoretic motion of a soft spherical nanoparticle in a nanopore filled with an electrolyte solution has been investigated using a continuum mathematical model. The model includes the Poisson-Nernst-Planck (PNP) equations for the ionic mass transport and the modified Stokes and Brinkman equations for the hydrodynamic fields outside and inside the porous membrane layer, respectively. The effects of the “softness” of the nanoparticle on its electrophoretic velocity along the axis of a nanopore are examined with changes in the ratio of the radius of the rigid core to the double layer thickness, the ratio of the thickness of the porous membrane layer to the radius of the rigid core, the friction coefficient of the porous membrane layer, the fixed charge inside the porous membrane layer of the particle and the ratio of the radius of the nanopore to that of the rigid core. The presence of the soft membrane layer significantly affects the particle electrophoretic mobility.     

Influence of boundary on the effect of double-layer polarization and the electrophoretic behavior of soft biocolloids

The electrophoresis of a soft particle comprising a rigid core and a charged porous membrane layer in a narrow space is modeled. This simulates, for example, the capillary electrophoresis of biocolloids such as cells and microorganisms, and biosensor types of device. We show that, in addition to the boundary effect, the effects of double-layer polarization (DLP) and the electroosmotic retardation flow can be significant, yielding interesting electrophoretic behaviors. For example, if the friction coefficient of the membrane layer and/or the boundary is large, then the DLP effect can be offset by the electroosmotic retardation flow, making the particle mobility to decrease with increasing double layer thickness, which is qualitatively consistent with many experimental observations in the literature, but has not been explained clearly in previous analyses. In addition, depending upon the thickness of double layer, the friction of the membrane layer of a particle can either retard or accelerate its movement, an interesting result which has not been reported previously. This work is the first attempt to show solid evidence for the influence of a boundary on the effect of DLP and the electrophoretic behavior of soft particles. The model proposed is verified by the experimental data in the literature. The results of numerical simulation provide valuable information for the design of bio-analytical apparatus such as nanopore-based sensing applications and for the interpretation of relevant experimental data.     

Influence of membrane layer properties on the electrophoretic behavior of a soft particle

The influence of the physical properties of the membrane layer of a soft particle, which comprises a rigid core and a porous membrane layer, on its electrophoretic behavior, is investigated. Because that influence was almost always neglected in the previous studies, the corresponding results can be unrealistic. The applicability of the model proposed is verified by the available theoretical and experimental results. The electrophoretic mobility of the particle under various conditions is simulated through varying the dielectric constant, the thickness, and the drag coefficient of the membrane layer, and the bulk ionic concentration. We show that under typical conditions, the deviation in the electrophoretic mobility arising from assuming that the dielectric constant of the membrane layer is the same as that of the bulk liquid phase can be in the order of 50%. In addition, the thicker the membrane layer and/or the higher the bulk ionic concentration, the larger the deviation. If the surface of the core of the particle is charged, as in the case of inorganic particles covered by an artificial membrane layer, the deviation at constant core surface potential is larger than that under other types of charged conditions. However, if the surface of the core is uncharged, as in the case of biocolloids, then that deviation becomes negligible. These findings are of fundamental significance to theoreticians in their analysis on the electrokinetic behaviors of soft particles, and to experimentalists in the interpretation of their data.     

Effect of stress-jump condition on electrophoretic behavior of a spherical dispersion of soft particles

The electrophoresis of a concentrated dispersion of soft particles, where a particle comprises a rigid core and an ion-penetrable membrane layer, is modeled theoretically, taking the effect of double-layer polarization into account. In particular, the influence of a stress-jump condition of the flow field at the membrane layer-liquid interface on the electrophoretic mobility of a particle is investigated. The type of particles considered mimic biocolloids, such as cells and microorganisms, and inorganic colloids covered by an artificial polymer layer such as surfactant molecules. A unit cell model is adopted to simulate the present spherical dispersion, and the governing equations and the associated boundary conditions are solved by a pseudo-spectral method based on Chebyshev polynomials. We show that while the stress-jump condition, characterized by a stress-jump coefficient, can have a significant influence on the mobility of a particle, the associated flow field is not influenced appreciably. Also, the influence of the stress-jump condition on the mobility of a particle depends largely on the nature of the membrane layer, characterized by its friction coefficient.     

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.     

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.     

Sedimentation of a composite particle in a spherical cavity

The boundary effect on the sedimentation of a colloidal particle is investigated theoretically by considering a composite sphere, which comprises a rigid core and an ion-penetrable membrane layer, in a spherical cavity. A pseudo-spectral method is adopted to solve the governing electrokinetic equations, and the influences of the key parameters on the sedimentation behavior of a particle are discussed. We show that both the qualitative and quantitative behaviors of a particle are influenced significantly by the presence of the membrane layer. For example, if the membrane layer is either free of fixed charge or positively charged and the surface potential of the rigid core is sufficiently high, the sedimentation velocity has a local minimum and the sedimentation potential has a local maximum as the thickness of the double layer varies. These local extrema are not observed when the membrane layer is negatively charged. If the double layer is thin, the influence of the fixed charge in the membrane layer on the sedimentation potential is inappreciable.     

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 a particle covered with an ion-penetrable membrane

The electrophoretic behavior of a planar particle covered by an ion-penetrable membrane, which simulates a biological entity, is investigated. We show that, in general, a point charge model will overestimate the electrophoretic mobility of a particle and the deviation increases with the increase in the concentration of fixed charge and with the decrease in the thickness of membrane layer. As in the case of a point charge model, the present model also predicts a local maximum in the absolute mobility as the thickness of membrane layer varies. If the sizes of counterions of various valences are the same, then the lower the valence of counterions, the larger the mobility, and the larger the counterions, the greater the mobility. The latter is consistent with the experimental observations in the literature. For the level of the concentration of fixed charge examined, the effect of coions on the mobility is negligible.     

Effect of ionic size on the deposition of charge-regulated particles to a charged surface

The deposition of charge-regulated particles to a rigid, planar charged surface is modeled theoretically, taking the effects of the excluded area arising from deposited particles and finite ionic sizes into account. Here, a particle comprises a rigid core and an ion-penetrable charged membrane layer, which represents a general type of particle. If the membrane layer has a negligible thickness, the particle simulates a regular inorganic particle, and if the membrane layer has a finite thickness, it simulates biocolloids such as cells. The results of numerical simulation reveal that the rate of particle deposition is faster under the following conditions: (1) lower potential of the planar surface, (2) thicker membrane, (3) higher counterion valance, (4) lower fixed charge density, (5) smaller counterions, (6) larger co-ions, (7) larger functional group, and (8) lower pH. Neglecting the sizes of ionic species may lead to an appreciable deviation in both the electrical repulsive force between particle and surface and the rate of deposition. Typical deviation for the former is approximately 20%, and that for the latter is approximately -75%.     

Effect of core charge density on the electrophoresis of a soft particle coated with polyelectrolyte layer

The electrophoresis of a charged soft particle with charged rigid core is considered under a weak imposed field condition. The rigid core of the soft particle is considered to have a finite dielectric permittivity and a fixed volume charge density. The electric potential distribution is determined by solving the Poisson-Boltzman equation out side the rigid core and a Poisson equation within the core along with continuity conditions on the core-shell interface. We have extended the analytic expression of Ohshima (Electrophoresis 27:526–533, 2006) for the electrophoretic mobility of a soft particle with a charged shell to include the effect of the volume charge density of the rigid core. Mobility based on the present expression matches exactly with the existing analytical solutions for a soft particle with an uncharged core. We have also made a comparison of our solution for mobility with an uncharged rigid core with the existing experimental results. The impact of the core charge density on the soft particle mobility is analyzed.     

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.     

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.     

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.     

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.     

Dynamic electrophoretic mobility of a concentrated dispersion of particles with a charge-regulated surface at arbitrary potential

The dynamic electrophoretic mobility of a concentrated dispersion of biocolloids such as cells and microorganisms is modeled theoretically. Here, a biological particle is simulated by a particle, the surface of which contains dissociable functional groups. The results derived provide basic theory for the quantification of the surface properties of a biocolloid through an electroacoustic device, which has the merit of making direct measurement on a concentrated dispersion without dilution. Two key parameters are defined to characterize the phenomenon under consideration: the first, A, is associated with the pH of the dispersion, and the second, B, is associated with the equilibrium constant of the dissociation reaction of the functional group. We show that if A is large and/or B is small, the surface potential is high, and the effect of double-layer polarization becomes significant. In this case the dynamic electrophoretic mobility may have a local maximum and a phase lead as the frequency of the applied electric field varies. Due to the hydrodynamic interaction between neighboring particles, the dynamic electrophoretic mobility decreases with the concentration of dispersion.     

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.