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

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

Diffusioosmosis and electroosmosis in a capillary slit with surface charge layers

This study analytically examines the steady diffusioosmotic and electroosmotic flows of an electrolyte solution in a fine capillary slit with each of its inside walls covered by a layer of adsorbed polyelectrolytes. In this solvent-permeable and ion-penetrable surface charge layer, idealized polyelectrolyte segments are assumed to distribute at a uniform density. The electric double layer and the surface charge layer may have arbitrary thicknesses relative to the gap width between the slit walls. The electrostatic potential distribution on a cross section of the slit is obtained by solving the linearized Poisson–Boltzmann equation, which applies to the case of low potentials or low fixed-charge densities. Explicit formulas for the fluid velocity profile due to the imposed electrolyte concentration gradient or electric field through the slit are derived as the solution of a modified Navier–Stokes/Brinkman equation. The results demonstrate that the structure of the surface charge layer can lead to an augmented or a diminished electrokinetic flow (even a reversal in direction of the flow) relative to that in a capillary with bare walls, depending on the characteristics of the capillary, of the surface charge layer, and of the electrolyte solution. For the diffusioosmotic flow with an induced electric field, competition between electroosmosis and chemiosmosis can result in more than one reversal in direction of the flow over a range of the Donnan potential of the adsorbed polyelectrolyte in the capillary.     

Electrokinetic flow in a capillary with a charge-regulating surface polymer layer

An analytical study of the steady electrokinetic flow in a long uniform capillary tube or slit is presented. The inside wall of the capillary is covered by a layer of adsorbed or covalently bound charge-regulating polymer in equilibrium with the ambient electrolyte solution. In this solvent-permeable and ion-penetrable surface polyelectrolyte layer, ionogenic functional groups and frictional segments are assumed to distribute at uniform densities. The electrical potential and space charge density distributions in the cross section of the capillary are obtained by solving the linearized Poisson-Boltzmann equation. The fluid velocity profile due to the application of an electric field and a pressure gradient through the capillary is obtained from the analytical solution of a modified Navier-Stokes/Brinkman equation. Explicit formulas for the electroosmotic velocity, the average fluid velocity and electric current density on the cross section, and the streaming potential in the capillary are also derived. The results demonstrate that the direction of the electroosmotic flow and the magnitudes of the fluid velocity and electric current density are dominated by the fixed charge density inside the surface polymer layer, which is determined by the regulation characteristics such as the dissociation equilibrium constants of the ionogenic functional groups in the surface layer and the concentration of the potential-determining ions in the bulk solution.     

Cell model of the direct current electrokinetics in salt-free concentrated suspensions: the role of boundary conditions

In this paper, a general electrokinetic theory for concentrated suspensions in salt-free media is derived. Our model predicts the electrical conductivity and the electrophoretic mobility of spherical particles in salt-free suspensions for arbitrary conditions regarding particle charge, volume fraction, counterion properties, and overlapping of double layers of adjacent particles. For brevity, hydrolysis effects and parasitic effects from dissolved carbon dioxide, which are present to some extent in more "realistic" salt-free suspensions, will not be addressed in this paper. These issues will be analyzed in a forthcoming extension. However, previous models are revised, and different sets of boundary conditions, frequently found in the literature, are extensively analyzed. Our results confirm the so-called counterion condensation effect and clearly display its influence on electrokinetic properties such as electrical conductivity and electrophoretic mobility for different theoretical conditions. We show that the electrophoretic mobility increases as particle charge increases for a given particle volume fraction until the charge region where counterion condensation takes place is attained, for the above-mentioned sets of boundary conditions. However, it decreases as particle volume fraction increases for a given particle charge. Instead, the electrical conductivity always increases with either particle charge for fixed particle volume fraction or volume fraction for fixed particle charge, whatever the set of boundary conditions previously referred. In addition, the influence of the electric permittivity of the particles on their electrokinetic properties in salt-free media is examined for those frames of boundary conditions.     

Electrokinetic transport through nanochannels

This article presents a numerical study of the electrokinetic transport phenomena (electroosmosis and electrophoresis) in a three-dimensional nanochannel with a circular cross-section. Due to the nanometer dimensions, the Boltzmann distribution of the ions is not valid in the nanochannels. Therefore, the conventional theories of electrokinetic flow through the microchannels such as Poisson-Boltzmann equation and Helmholtz-Smoluchowski slip velocity approach are no longer applicable. In the current study, a set of coupled partial differential equations including Poisson-Nernst-Plank equation, Navier-Stokes, and continuity equations is solved to find the electric potential field, ionic concentration field, and the velocity field in the three-dimensional nanochannel. The effects of surface electric charge and the radius of nanochannel on the electric potential, liquid flow, and ionic transport are investigated. Unlike the microchannels, the electric potential field, ionic concentration field, and velocity field are strongly size-dependent in nanochannels. The electric potential gradient along the nanochannel also depends on the surface electric charge of the nanochannel. More counter ions than the coions are transported through the nanochannel. The ionic concentration enrichment at the entrance and the exit of the nanochannel is completely evident from the simulation results. The study also shows that the flow velocity in the nanochannel is higher when the surface electric charge is stronger or the radius of the nanochannel is larger.     

Dielectric response of a concentrated colloidal suspension in a salt-free medium

In this paper the complex dielectric constant of a concentrated colloidal suspension in a salt-free medium is theoretically evaluated using a cell model approximation. To our knowledge this is the first cell model in the literature addressing the dielectric response of a salt-free concentrated suspension. For this reason, we extensively study the influence of all the parameters relevant for such a dielectric response: the particle surface charge, radius, and volume fraction, the counterion properties, and the frequency of the applied electric field (subgigahertz range). Our results display the so-called counterion condensation effect for high particle charge, previously described in the literature for the electrophoretic mobility, and also the relaxation processes occurring in a wide frequency range and their consequences on the complex electric dipole moment induced on the particles by the oscillating electric field. As we already pointed out in a recent paper regarding the dynamic electrophoretic mobility of a colloidal particle in a salt-free concentrated suspension, the competition between these relaxation processes is decisive for the dielectric response throughout the frequency range of interest. Finally, we examine the dielectric response of highly charged particles in more depth, because some singular electrokinetic behaviors of salt-free suspensions have been reported for such cases that have not been predicted for salt-containing suspensions.     

Mobilization of electroosmotic flow markers in capillary zone electrophoresis

UV-absorbing neutral substances are commonly used as markers of mean electroosmotic flow in capillary electrophoresis for their zero electrophoretic mobility in an electric field. However, some of these markers can interact with background electrolyte components and migrate at a different velocity than the electroosmotic flow. Thus, we tested 11 markers primarily varying in their degree of methylation and type of central atom in combination with five background electrolyte cations differing in their ionic radii and surface charge density, measuring the relative electrophoretic mobility using thiourea as a reference marker. Our results from this set of experiments showed some general trends in the mobilization of the markers based on the effects of marker structure and type of background electrolyte cation on the relative electrophoretic mobility. As an example, the effects of an inadequate choice of marker on analyte identification were illustrated in the electrophoretic separation of glucosinolates. Therefore, our findings may help electrophoretists appropriately select electroosmotic flow markers for various electrophoretic systems.     

The application of spline wavelet transform to the determination of electroosmotic mobility in capillary electrophoresis

The rule of current change was studied during capillary electrophoresis (CE) separation process while the conductivity of the sample solution was different from that of the buffer. Using a quadratic spline wavelet of compact support, the wavelet transforms (WTs) of capillary electrophoretic currents were performed. The time corresponding to the maximum of WT coefficients was chosen as the time of current inflection to calculate electroosmotic mobility. The proposed method was suitable for different CE modes, including capillary zone electrophoresis, nonaqueous CE and micellar electrokinetic chromatography. Compared with the neutral marker method, the relative errors of the developed method for the determination of electroosmotic mobility were all below 2.5%.     

Scaling of electrokinetic transport in nanometer channels

Electrokinetic transport is a popular transport mechanism used in nanofluidic systems, and understanding its scaling behavior is important for the design and optimization of nanofluidic devices. In this article, we report on the scaling of electroosmotic flow and ionic conductivity in positively charged slit nanochannels by using continuum and atomistic simulations. The effects of confinement and surface charge are discussed in detail. In particular, we found that the viscosity of the interfacial water increases substantially as the surface charge density increases and the electrophoretic mobility of the interfacial ions decreases. We show that such effects can influence the scaling of the electrokinetic transport in confined nanochannels significantly.     

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.     

The Electrophoretic Mobility and Electric Conductivity of a Concentrated Suspension of Colloidal Spheres with Arbitrary Double-Layer Thickness

The electrophoresis in a monodisperse suspension of dielectric spheres with an arbitrary thickness of the electric double layers is analytically studied. The effects of particle interactions are taken into account by employing a unit cell model, and the overlap of the double layers of adjacent particles is allowed. The electrokinetic equations, which govern the ionic concentration distributions, the electric potential profile, and the fluid flow field in the electrolyte solution surrounding the charged sphere in a unit cell, are linearized assuming that the system is only slightly distorted from equilibrium. Using a perturbation method, these linearized equations are solved with the surface charge density (or zeta potential) of the particle as the small perturbation parameter. Analytical expressions for the electrophoretic mobility of the colloidal sphere in closed form correct to O(zeta) are obtained. Based on the solution of the electrokinetic equations in a cell, a closed-form formula for the electric conductivity of the suspension up to O(zeta(2)) is derived from the average electric current density. Comparisons of the results of the cell model with different conditions at the outer boundary of the cell are made for both the electrophoretic mobility and the electric conductivity. Copyright 2001 Academic Press.     

Analysis of effect of electrolyte types on electrokinetic energy conversion in nanoscale capillaries

An analytical study on the effect of electrolyte types on the electrokinetic energy conversion is presented using nanoscale cylindrical capillary, which is either positively or negatively charged. The sign of surface charge determines the role and concentration magnitude of ions in the capillary and the energy conversion performance. Our study shows that the electrokinetic energy conversion performance (maximum efficiency, pressure rise and streaming potential) are approximately identical for 1:1 (KCl), 2:1 (CaCl₂) and 3:1 (LaCl₃) electrolytes when capillary is positively charged. For negatively charged capillary, energy conversion performance degrades significantly with the increase of counter‐ion valence. For both positively and negatively charged capillaries, higher maximum efficiency can be resulted in low bulk concentration and surface charge density regimes. However, high maximum pressure rise generation for the pumping is found in the low bulk concentration and high surface charge density regimes. For the electric power generation, higher maximum streaming potential is found when both bulk concentration and surface charge density are low.     

Electroosmotic flow in a capillary annulus with high zeta potentials

The electroosmotic flow through an annulus is analyzed under the situation when the two cylindrical walls carry high zeta potentials. The analytical solutions for the electric potential profile and the electroosmotic flow field in the annulus are obtained by solving the Poisson-Boltzmann equation and the Stokes equation under an analytical scheme for the hyperbolic sine function. A mathematical expression for the average electroosmotic velocity is derived in a fashion similar to the Smoluchowski equation. Hence, a correction formula is introduced to modify the Smoluchowski equation, taking into account contributions due to the finite thickness of the electric double layer (EDL) and the geometry ratio-dependent correction. Specifically, under a circumstance when the two annular walls are oppositely charged, the flow direction can be determined from the sign of such correction formula, and there exists a zero-velocity plane inside the annulus. With the assumption of large electrokinetic diameters, the location of the zero-velocity plane can be estimated from the analytical expression for the velocity distribution. In addition, the characteristics of the electroosmotic flow through the annulus are discussed under the influences of the EDL parameters and geometric ratio of the inner radius to the outer radius of the annulus.     

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.     

Non-equilibrium electric surface phenomena

Non-equilibrium aspects of traditional electrokinetic phenomena (electrophoresis, electroosmosis, streaming potential, sedimentation potential), electrostatic interaction of particles and new electrokinetic phenomena are considered. The significance of non-equilibrium electric surface phenomena for many major areas of modern colloid science (characterization of colloids, membrane science, transport phenomena and separation, particle interaction and coagulation) is established.The study of non-equilibrium electric surface phenomena is connected with the validation of the standard electrokinetic model (SEM), the development of a non-standard model and the development of an extensive programme of disperse system characterization based on integrated electrokinetic investigations. Experimental and theoretical studies of systems with a smooth, non-porous impermeable surface (mica in Anderson's experiments, and quartz microcapillaries with a molecule-smooth surface in Churaev's experiments) have shown that usually there are no significant difficulties in interpreting electrokinetic investigations despite the possible anomaly in the water structure near the surface and the possibility of maximum shear stress (yield stress), i.e. the anomalous viscosity and decreased dissolving power with respect to ions. However, systems which do not satisfy the conditions of the SEM are widely distributed, owing to the porosity, roughness or permeability of the boundary layer of the surface of the solid body which simultaneously belongs to the solid and liquid phases. In this layer, enclosed between the outer Helmholtz plane and the slipping plane, the motion of the liquid strongly slows down and the tangential flow of ions is characterized purely by the mobility which is close to the normal. Thus, a general property of a non-standard electrokinetic model is the presence of an anomalous (additional) surface conductivity in excess of the surface conductivity determined according to Bikerman's equation based on the ζ -potential alone.Confidence in modelling the electrokinetic phenomena has grown with the development of methods for modifying the surface such that its properties approach those of the SEM (Bijsterbosch and co-workers; Saville and co-workers).Extension of the particle characterization concept requires the measurement of both the mobile charge and the electrokinetic charge and from this an estimate of the thickness of the additional conductivity zone can be made. With the additional measurement of a titratable charge, it is possible to estimate the ion distribution between the dense and diffuse parts of the double layer (DL) and to estimate the decreased mobility of ions in the Stern layer or in the immobilized part of the DL.Quantitative laws governing the interaction of particles and corresponding to the non-standard model substantially differ from the traditional laws described by the DVLO theory as applied to the SEM. This is also true for adsorption properties which are characterized without sufficient reason by means of the ζ-potential. Therefore both the development of models of interaction and adsorption of ions, allowing for the non-standard electrokinetic model, and the extension of the particle characterization programme to integrated investigations of electric surface phenomena are required.Further generalization of the theory of electrokinetic phenomena is achieved. In addition to the surface charge another variety of surface force can be the origin of the electrokinetic phenomena.     

A numerical study on ion concentration polarization and electric circuit performance of an electrokinetic battery

Ion concentration polarization (ICP) imposes remarkable adverse effects on the energy conversion performance of the pressure-driven electrokinetic (EK) flows through a capillary system that can be equivalently treated as a battery. An optimized dimensionless numerical method is proposed in this study to investigate the causes and the effects of the ICP. Results show that remarkable ICP phenomena are induced under certain conditions such as high applied pressure, high surface charge density, and small inversed Debye length at dimensionless values of 6000, ‒10, and 0.5. Meanwhile, different factors influence the ICP and the corresponding electric properties in different ways. Particularly for the overall electric resistance, the applied pressure and the surface charge density mainly affect the variation amplitude and the level of the overall electric resistance when varying the output electric potential, respectively. Differently, the Debye length affects the overall electric resistance in both aspects. Ultimately, the induced ICP leads to significant nonlinear current–potential curves.     

Electroosmotic velocity in an array of parallel soft cylinders in a salt-free medium

A theory of electroosmosis in an array of parallel soft cylinders (i.e. polyelectrolyte-coated cylinders) in a salt-free medium is presented. It is shown that there is a certain critical value of the particle charge and that if the particle charge is greater than the critical value, then the electroosmotic velocity becomes constant independent of the particle charge due to the counterion condensation effects, as in the case of other electrokinetic phenomena in salt-free media.     

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.