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.     

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.     

Electrophoresis of a concentrated dispersion of spherical particles covered by an ion-penetrable membrane layer

The electrophoretic behavior of a concentrated dispersion of soft spherical particles is investigated theoretically, taking the effects of double-layer overlapping and double-layer polarization into account. Here, a particle comprises a rigid core and an ion-penetrable layer containing fixed charge, which mimics biocolloids and particles covered by artificial membrane layers. A cell model is adopted to simulate the system under consideration, and a pseudo-spectral method based on Chebyshev polynomials is chosen for the resolution of the governing electrokinetic equations. The influence of the key parameters, including the thickness of the double layer, the concentration of particles, the surface potential of the rigid core of a particle, and the thickness, the amount of fixed charge, and the friction coefficient of the membrane layer of a particle on the electrophoretic behavior of the system under consideration is discussed. We show that while the result for the case of a dispersion containing rigid particles can be recovered as the limiting case of a dispersion containing soft particles, qualitative behaviors that are not present in the former are observed in the latter.     

Electrophoresis of a biocolloid covered with a cation-absorptive membrane

The electrophoretic behavior of a biocolloid covered with a charged membrane is theoretically analyzed in the present study. Here, the influences of nonuniformly distributed fixed groups, absorption of cations by fixed original functional groups, variation in dielectric constant in the electrophoretic system, and ionic sizes are considered. The results of numerical simulation suggest that a larger absolute value of the electrophoric mobility of biocolloids could be generated by larger membrane electricity. The absolute value of the electrophoric mobility for the nonlinear distribution of the fixed groups is larger than that for the linear distribution of the fixed groups. The absolute value of the electrophoric mobility increases with (1) the concentration of total fixed groups, (2) the cation-absorption equilibrium constant, (3) the nonuniform feature index for functional-groups distribution, (4) the dielectric constants of the inner uncharged membrane zone for only mobile cationic charge and for both mobile cationic and anionic charge, and (5) the effective size of the cations. An increase in the absolute value of the electrophoric mobility can also be resulted from a decrease in the following parameters: (1) the friction coefficient of the biocolloidal membrane phase, (2) the membrane thickness, (3) the dielectric constant of space for all charge and of outer uncharged membrane zone, (4) the effective sizes of anions and fixed groups, and (5) the number of cations and the fixed original functional groups involved in the formation of a cation-functional group complex.     

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.     

A new generalization of the standard electrokinetic model

We present a new generalization of the standard electrokinetic model based on the assumption that there is a thin layer surrounding the suspended particle where the equilibrium ion density is not determined by the Gouy-Chapman distribution, while the standard model applies outside this layer. Our approach differs from existing models in that we consider that the surface layer is made both of free ions (mostly counterions) and of the fixed ions that constitute the charge of the particle. Furthermore, the free ion density is determined by appropriate boundary conditions without considering any adsorption isotherms. Finally, the fluid is allowed to freely flow inside the layer, only hindered by the presence of the fixed charges and the adhesion condition on the surface of the particle. We show that this generalization leads to results that qualitatively differ from those obtained using existing models: instead of always decreasing, the electrophoretic mobility can actually increase with the anomalous surface conductivity. This could make it possible to use our model for the interpretation of a broader set of experimental data, including those cases when the measured mobility is higher than predicted by the standard model.     

Complexation of ferric oxide particles with pectins of different charge density

The effect of polyelectrolyte charge density on the electrical properties and stability of suspensions of oppositely charged oxide particles is followed by means of electro-optics and electrophoresis. Variations in the electro-optical effect and the electrophoretic mobility are examined at conditions where fully ionized pectins of different charge density adsorb onto particles with ionizable surfaces. The charge neutralization point coincides with the maximum of particle aggregation in all suspensions. We find that the concentration of polyelectrolyte, needed to neutralize the particle charge, decreases with increasing charge density of the pectin. The most highly charged pectin presents an exception to this order, which is explained with a reduction of the effective charge density of this pectin due to condensation of counterions. The presence of condensed counterions, remaining bound to the pectin during its adsorption on the particle surface, is proved by investigation of the frequency behavior of the electro-optical effect at charge reversal of the particle surface.     

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%.     

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.     

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.     

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 a cylindrical colloidal particle in a salt-free medium

Approximate expressions are derived for the electrophoretic mobility of dilute cylindrical colloidal particles in a salt-free medium containing only counterions. The cylinder is assumed to be infinitely long. It is shown that as in the case of a spherical particle, there is a certain critical value of the particle surface charge separating two cases. When the particle surface charge is lower than the critical value (case 1), the electrophoretic mobility increases with increasing particle surface charge per unit length. When the particle surface charge is higher than the critical value (case 2), the mobility becomes constant (for a cylinder in a transverse field) or the increase in the electrophoretic mobility with the particle surface charge becomes suppressed (for a cylinder in a tangential field). These phenomena are caused by the effect of counterion condensation in the vicinity of the particle surface. The critical value of the particle charge is essentially independent of the particle volume fraction phi for the dilute case, unlike the case of a sphere, in which case the critical charge value is proportional to ln(1/phi).     

Electrokinetic transport of a spherical gel-layer model particle: inclusion of charge regulation and application to polystyrene sulfonate

An electrokinetic gel-layer model of a spherical, highly charged colloid particle developed previously [S. Allison, J. Colloid Interface Sci. 277 (2004) 248], is extended in several ways. The charge of the particle is assumed to arise from deprotonation of acidic groups that are present, in uniform concentration, over a portion (or all) of the gel layer. Free energy considerations coupled with Poisson-Boltzmann theory are used to estimate how the local electrostatic environment of a charged gel layer alters the local pK(a) of the acidic groups. This modulation of the charge of the colloidal particle, or "charge regulation," can be significant even for colloidal particles with strongly acidic groups at moderate pH if the ambient salt concentration is low. The methodology is applied to the viscosity and electrophoretic mobility data of a particular polystyrene sulfonate latex over a broad range of monovalent salt (NaCl) concentration [M.J. Garcia-Salinas, F.J. de las Nieves, Langmuir 16 (2000) 715]. The experimental data can be accounted for by a gel layer model that decreases in thickness, but does not vanish, as the salt concentration is increased. Viscosity data provides valuable information about the degree of solvation of the colloidal particle and the thickness of the gel layer. The mobility data is best explained by a model in which only the outermost portion of the gel layer is charged. Charge regulation is significant at a monovalent salt concentration of 3 x 10(-3) mol/l and increases as the salt concentration decreases.     

Counterion condensation in hydrosols of single-crystalline detonation nanodiamond particles obtained by air annealing of their agglomerates

The pattern of previously recorded dependences of the specific surface charge and electrophoretic mobility of monodisperse detonation nanodiamond particles on pH of aqueous KCl solutions suggests that counterions are condensed on the particle surface. Counterion condensation is considered in terms of the Levin model, and the experimental ratios between the densities of the electrokinetic and surface charges of dispersed particles, as well as the fractions of condensed counterions, are calculated as depending on pH and KCl concentration in nanodiamond hydrosols. The obtained dependences lead to the conclusion that counterion condensation on the surface of detonation nanodiamond particles does indeed take place.     

Numerical study of colloidal suspensions of soft spherical particles using the network method. 1. DC electrophoretic mobility

The electrophoretic mobility of a spherical particle coated with a uniformly charged permeable membrane and suspended in a general electrolyte solution is calculated numerically. The network simulation method used makes it possible to solve the problem without any restrictions on the values of the parameters such as the membrane thickness, fixed charge density in the membrane, viscous drag in the membrane, number and valence of the ionic species, and electrolyte concentration. The theoretical model used is similar to the one presented by Ohshima (H. Ohshima, J. Colloid Interface Sci. 228 (2000) 190), except for 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. This inclusion is theoretically proven in the limiting case of a nonconducting suspending medium, in which the equation system can be analytically solved. The results obtained coincide with existing analytical expressions when the electrolyte concentration is high, the membrane is thick, and its resistance to the fluid flow is high.     

dc Electrokinetics for spherical particles in salt-free concentrated suspensions including ion size effects

We study the electrophoretic mobility of spherical particles and the electrical conductivity in salt-free concentrated suspensions including finite ion size effects. An ideal salt-free suspension is composed of just charged colloidal particles and the added counterions that counterbalance their surface charge. In a very recent paper [Roa et al., Phys. Chem. Chem. Phys., 2011, 13, 3960-3968] we presented a model for the equilibrium electric double layer for this kind of suspensions considering the size of the counterions, and now we extend this work to analyze the response of the suspension under a static external electric field. The numerical results show the high importance of such corrections for moderate to high particle charges, especially when a region of closest approach of the counterions to the particle surface is considered. The present work sets the basis for further theoretical models with finite ion size corrections, concerning particularly the ac electrokinetics and rheology of such systems.     

Nonlinear effects on electrokinetics of a highly charged porous sphere

The motivation of the present study is to provide a correct estimate of the electrophoretic mobility of a charged porous particle for wide-range electrokinetic parameters, such as particle charge density, permeability, and Debye length. Based on the Nernst–Planck equation, which takes into account the external electric field and fluid convection on ion transport, we have estimated the mobility of the particle by establishing a force balance. We have validated our results with the linear model due to Hermans and Fujita (K Nederl Akad Wet Proc Ser B 58:182–187, 1955) and the computed solution based on perturbation of the Poisson–Boltzmann model as obtained by Hsu and Lee (J Colloid Interface Sci 390:85–95, 2013). For the case of thin double layer, our computed results agree with the linear model even for large values of charge density of the particle. The linear model overpredicts our computed solution for mobility when the thick Debye layer is considered. However, a large discrepancy of the present model from the results based on the perturbation of the Boltzmann model is observed for all the cases considered. We have analyzed the double-layer polarization and counterion condensation through the distribution of counterions, net charge density, and the effective charge density of the particle.     

Electrokinetic Properties of Fuzzy Colloidal Particles

The presence of a thin polymer layer on the surface of a colloidal particle can have a profound effect on its electrophoretic mobility. The model developed here treats the hydrodynamics of the polymer layer as a distribution of Stokes resistance centers within a thin diffuse layer; fixed charge may reside on the surface of the particle core or throughout the layer. The theory is semianalytical in that asymptotic methods are used to simplify the equations but several integrals must be evaluated numerically. Special attention is paid to the effects of polarization and relaxation. It is shown that distributing immobile charge throughout the layer produces a response where the particle's mobility exceeds that found when the same amount of charge is spread uniformly over the surface of the rigid core. Increasing the drag due to the fuzzy layer always diminishes the mobility. In either case, the hydrodynamic permeability of the layer has a strong influence on particle movement. Results are also given for the dipole coefficient in the expression for the conductivity of a dilute suspension. Copyright 2000 Academic Press.     

Electrophoresis of a spherical dispersion of polyelectrolytes in a salt-free solution

The electrophoretic behavior of a spherical dispersion of polyelectrolytes of arbitrary concentration is analyzed theoretically under a salt-free condition, that is, the liquid phase contains only counterions which come from the dissociation of the functional groups of polyelectrolytes. We show that, in general, the surface potential of a polyelectrolyte increases nonlinearly with its surface charge. A linear relation exists between them, however, when the latter is sufficiently small; and the more dilute the concentration of polyelectrolytes, the broader the range in which they are linearly correlated. If the amount of surface charge is sufficiently large, counterion condensation occurs, and the rate of increase of surface potential as the amount of surface charge increases declined. Also, it leads to an inverse in the perturbed potential near the surface of a polyelectrolyte, and its mobility decreases accordingly. For a fixed amount of surface charge, the lower the concentration of polyelectrolytes and/or the lower the valence of counterions, the higher the surface potential. The qualitative behavior of the mobility of a polyelectrolyte as the amount of its surface charge varies is similar to that of its surface charge.