Standard Electrokinetic Model Research Articles

Electrorotation of leaky-dielectric and conducting microspheres in asymmetric electrolytes and angular velocity reversal.

We consider the central problem of polarizable and leaky-dielectric uncharged spherical particle freely suspended in an unbounded nonsymmetric binary electrolyte, which is forced by an ambient time-harmonic uniform electric field. Under the assumption of a "weak field," we employ the linearized standard electrokinetic model of binary electrolytes to account for such anion/cation asymmetry. A simplified generalized asymmetric dipole-term approximation, valid for a dielectric/conducting microsphere, is analytically derived for an arbitrary Debye scale and for any mismatch between ion diffusivities and valances. A two-peak unified dispersion spectrum covering all range of practical frequencies (KHz to MHz), is found for the case of a rotating electric field (ROT). The angular velocity of a free polarized particle is composed of dielectrophoretic contribution, resulting from the electrical torque (dipole term) as well as from the induced electroosmotic (ICEO) flow field. The two effects usually act in opposite directions. Under ROT excitation, we obtain a cofield rotation at high frequencies (MHz) and a counter-field behavior at low frequencies (KHz). The low-frequency dispersion is generally governed by electric double-layer charging and the high frequency by a Maxwell-Wagner relaxation process. ICEO generally dominates the low-frequency cofield response; however, it can be shown that depending on the electrolyte asymmetry, yet another dielectrophoretic related switching (reversal) point might exist. Furthermore, for large frequencies and depending on the complex permittivity ratio between the particle and electrolyte, we find a second switching point. Explicit expressions for the above two frequency reversal values are obtained in terms of the problem physical parameters and are compared against experimental results. Finally, we provide an analytical solution for the ROT ICEO velocity field of a microsphere as a function of electrolyte asymmetry and Debye length and compare it with numerical simulations.

Transport Properties in Nanochannels: Ionic Size-, Permittivity-, and Viscosity-Related Effects

The transport properties in nanochannels are examined using a modified electrokinetic model (MEM) that takes into account the finite ion size by modeling the hydrated ions as charged dielectric spheres so that the electrolyte solution becomes a nonhomogeneous fluid characterized by permittivity and viscosity values that are both functions of the local ionic concentrations. We show that the simultaneous consideration of ionic size, permittivity, and viscosity effects always improve on the predictions of the standard electrokinetic model for point-like ions (SEM), with corrections that are significant in the nanofluidic system’s parameter range. The results reported herein are expected to provide a deeper understanding of the ion transport and fluid flow in nanofluidic devices.

Ionic size, permittivity, and viscosity-related effects on the electrophoretic mobility: A modified electrokinetic model

The standard electrokinetic model (SEM) is modified by considering hydrated ions as charged dielectric spheres so that in the modified model (MEM) the electrolyte solution becomes a non-homogeneous fluid with permittivity and viscosity values that are both functions of the local ionic concentrations.

Numerical Solution of the Electrokinetic Equations for Multi-ionic Electrolytes Including Different Ionic Size Related Effects.

One of the main assumptions of the standard electrokinetic model is that ions behave as point-like entities. In a previous work (López-García, et al., 2015) we removed this assumption and analyzed the influence of finite ionic size on the dielectric and electrokinetic properties of colloidal suspensions using both the Bikerman and the Carnahan–Starling equations for the steric interactions. It was shown that these interactions improved upon the standard model predictions so that the surface potential, electrophoretic mobility, and the conductivity and permittivity increment values were increased. In the present study, we extend our preceding works to systems made of three or more ionic species with different ionic sizes. Under these conditions, the Bikerman and Carnahan–Starling expressions cease to be valid since they were deduced for single-size spheres. Fortunately, the Carnahan–Starling expression has been extended to mixtures of spheres of unequal size, namely the “Boublik–Mansoori–Carnahan–Starling–Leland” (BMCSL) equation of state, making it possible to analyze the most general case. It is shown that the BMCSL expression leads to results that differ qualitatively and quantitatively from the standard electrokinetic model.

Ionic coupling effects in dynamic electrophoresis and electric permittivity of aqueous concentrated suspensions

In the recent past the standard electrokinetic model formulated for a concentrated suspension subjected to a static electric field has been checked in detail against more sophisticated models which correctly compute effects associated to the so-called added or released counterions (i.e., those compensating the surface charge of the particles), or to realistic descriptions of the chemistry of the solution. Important, as it was evidenced in dc fields, is to explore the consequences of neglecting the above-mentioned features in the even more significant case of ac fields, whereby details of the electrokinetic response are observed which go unnoticed at constant fields. In the present contribution, a general ac electrokinetic cell model for a concentrated suspension in an arbitrary electrolyte solution is developed that includes all the effects mentioned. Attention is paid to the frequency spectra of the dynamic electrophoretic mobility and the dielectric dispersion of concentrated suspensions. Different coupling situations are considered between the released counterions and the ions from the background salt solution (the only ones accounted for within the standard model), that have been shown to be responsible for some important deviations in electrokinetic predictions when a comparison is made against standard results. Differences between them are mainly noticeable at high field frequencies regarding the Maxwell-Wagner counterion condensate relaxation, but also have been found at lower frequencies, when Maxwell-Wagner-O’Konski and alpha- or concentration polarization relaxations are concerned, due to overall changes in: released counterion mobilities, screening effects on particles charge or ionic concentration profiles in the double layers. The complementarity of dynamic mobility and dielectric dispersion is well appreciated in distinguishing the different situations. In the paper, the way in which these general features are affected by changes in ionic strength, particle size and charge, or solids concentration, are discussed. It is expected that this contribution can provide a better understanding of the electrokinetic response of suspensions of nanoparticles in realistic aqueous media due to their relevance for many applications in industry and biotechnology.

Differential capacitance of the diffuse double layer at electrode-electrolyte interfaces considering ions as dielectric spheres: Part I. Binary electrolyte solutions

A full theoretical account of the differential capacitance of the diffuse part of the electric double layer at electrode-electrolyte solution interfaces is presented. It builds upon the standard electrokinetic model adding all the additional effects related to the finite ionic size. This includes steric interactions among ions by means of either the Bikerman or Carnahan-Starling expressions and all the permittivity related effects that arise when ions are represented as dielectric spheres. These include the solution permittivity dependence on the local ionic concentration, calculated by means of the Maxwell mixture formula, and two additional forces acting on the ions, namely the Born and the dielectrophoretic forces that depend on the permittivity and the electric field gradients, respectively. The obtained results show that the diffuse double layer behavior is sufficient to qualitatively account for the observed differential capacitance dependence on the electrode voltage. Moreover, when combined with an inner layer capacitance and using the Carnahan-Starling expression, a remarkably good quantitative agreement is achieved.

Hydrogel charge regulation and electrolyte ion-concentration perturbations in nanoparticle gel electrophoresis

Gel electrophoresis of spherical nanoparticles (NPs) is studied using an electrokinetic model that couples the ion conservation equations to the Poisson and fluid momentum equations, thus including the so-called polarization and relaxation processes. This model is therefore the charged gel electrophoresis analogue of the well-known O’Brien and White solution of the standard electrokinetic model for free-solution electrophoresis. Results are provided for the small NPs (size around 10 nm) to which gel electrophoresis is relevant, because particles must be small enough to permeate the gel: these include the particle drag coefficient (or Brownian diffusivity), which is subject to hydrodynamic screening and electroviscous effects, and the electrophoretic mobility, which is subject to nonlinear electrostatic and charge polarization influences. Also addressed are the influences of charge-regulating gels and the accompanying particle-induced immobile charge-density perturbations. Ion-concentration perturbations attenuate the electrophoretic mobility and enhance the drag coefficient according to the particle charge and the mobility of the most abundant counterion. However, dynamic regulation of the hydrogel charge—termed the secondary immobile charge-density perturbation—has a negligible influence on the particle mobility, and may therefore be neglected for most practical purposes.

Influence of steric interactions on the dielectric and electrokinetic properties in colloidal suspensions

One of the main assumptions of the standard electrokinetic model is that ions behave as point like entities. In this work we remove this assumption and analyze the main consequences of finite ionic size on the dielectric and electrokinetic properties of colloidal suspensions. We represent the steric interactions by means of the Bikerman and the Carnahan–Starling equations and solve numerically the standard linearized electrokinetic equations in the stationary and the frequency domains, for surface charge density and electrolyte solution concentration values typically encountered in colloidal suspensions. In all cases the steric interactions improve upon the predictions of the standard model since the surface potential, the electrophoretic mobility, and the conductivity and permittivity increments increase. However, the corrections introduced by the Bikerman equation are generally small: less than 10% as compared to the standard model. On the contrary, the Carnahan–Starling equation leads to corrections to the surface potential versus surface charge and the electrophoretic mobility values that easily surpass 10% and can attain values as high as 50%. Corrections to the conductivity and permittivity increments are smaller but still non negligible.

General electrokinetic model for concentrated suspensions in aqueous electrolyte solutions: Electrophoretic mobility and electrical conductivity in static electric fields

In recent years different electrokinetic cell models for concentrated colloidal suspensions in aqueous electrolyte solutions have been developed. They share some of its premises with the standard electrokinetic model for dilute colloidal suspensions, in particular, neglecting both the specific role of the so-called added counterions (i.e., those released by the particles to the solution as they get charged), and the realistic chemistry of the aqueous solution on such electrokinetic phenomena as electrophoresis and electrical conductivity. These assumptions, while having been accepted for dilute conditions (volume fractions of solids well below 1%, say), are now questioned when dealing with concentrated suspensions. In this work, we present a general electrokinetic cell model for such kind of systems, including the mentioned effects, and we also carry out a comparative study with the standard treatment (the standard solution only contains the ions that one purposely adds, without ionic contributions from particle charging or water chemistry). We also consider an intermediate model that neglects the realistic aqueous chemistry of the solution but accounts for the correct contribution of the added counterions. The results show the limits of applicability of the classical assumptions and allow one to better understand the relative role of the added counterions and ions stemming from the electrolyte in a realistic aqueous solution, on electrokinetic properties. For example, at low salt concentrations the realistic effects of the aqueous solution are the dominant ones, while as salt concentration is increased, it is this that progressively takes the control of the electrokinetic response for low to moderate volume fractions. As expected, if the solids concentration is high enough the added counterions will play the dominant role (more important the higher the particle surface charge), no matter the salt concentration if it is not too high. We hope this work can help in setting up the real limits of applicability of the standard cell model for concentrated suspensions by a quantitative analysis of the different effects that have been classically disregarded, showing that in many cases they can be determinant to get rigorous predictions.

Electrophoresis of a Spherical Polyelectrolyte-Grafted Colloid in Monovalent Salt Solutions: Comparison of Molecular Dynamics Simulations with Theory and Numerical Calculations

We study the electrophoresis of a polyelectrolyte-grafted colloid immersed in monovalent salt solution by means of explicit-ion molecular dynamics simulations including long-ranged hydrodynamic interactions via a lattice-Boltzmann fluid. We expand on our previous work on the subject by investigating the effect of charges and the salt concentration on both net-neutral and net-charged composites. We show that the dominant role of the polyelectrolytes at higher salt concentrations prevents a net-negative soft colloid to reverse mobility with respect to net charge except at extremely high net charges. We explain all our simulation results by examining in detail the ion and monomer density profiles around the colloid and the ratio of the Debye length to the brush height. Our simulation results show good agreement with both approximate analytic expressions and numerical solutions based on an extended standard electrokinetic model for polymer-grafted spherical colloids.

Computing the Electrophoretic Mobility of Large Spherical Colloids by Combining Explicit Ion Simulations with the Standard Electrokinetic Model

The electrophoretic mobility of large spherical colloids in different salt solutions of varying valency and concentration is studied via a combination approach of numerically solving the standard electrokinetic model with a ζ potential that has been obtained from explicit ion simulations of the restricted primitive model, thus going beyond the standard mean-field treatment. We compare our theoretical mobility curves to two distinct sets of experimental results and obtain good agreement for monovalent and divalent salt solutions. For the case of the trivalent La(3+) salt, the experimentally obtained mobility reversal at high ionic strengths can be obtained only by adding an additional attractive interaction of 4k(B)T to the potential between the colloid and La(3+), hinting at the presence of a nonelectrostatic binding term for this ion. It is also shown that, contrary to intuition, charge inversion does not necessarily result in mobility reversal.

Electrophoretic mobility and charge inversion of a colloidal particle studied by single-colloid electrophoresis and molecular dynamics simulations

Optical Tweezers are employed to study the electrophoretic and the electroosmotic motion of a single colloid immersed in electrolyte solutions of ion concentrations between 10(-5) and 1 mol/l and of different valencies (KCl, CaCl(2), LaCl(3)). The measured particle mobility in monovalent salt is found to be in agreement with computations combining primitive model molecular dynamics simulations of the ionic double layer with the standard electrokinetic model. Mobility reversal of a single colloid-for the first time-is observed in the presence of trivalent ions (LaCl(3)) at ionic strengths larger than 10(-2) mol/l. In this case, our numerical model is in a quantitative agreement with the experiment only when ion specific attractive forces are added to the primitive model, demonstrating that at low colloidal charge densities, ion correlation effects alone do not suffice to produce mobility reversal.

Electrophoretic mobility of latex nanospheres in electrolytes: Experimental challenges

The electrophoretic mobility of sulfate latex nanospheres (radius 300 ± 10 nm) was measured as a function of ionic strength for different salts. The results were obtained from two similar instruments (Malvern ZetaSizer 3000 HSa and Malvern ZetaSizer Nano) using the same dispersions, in the same conditions. The values predicted from the standard electrokinetic model for constant surface charge were in good agreement with the data over a large range of ionic strength. The influence of the protocol used to fill the cells appears to be of importance between 1–10 mM of added monovalent salt. There, the capillary wall properties seem to influence the electrophoretic measurements, even at fast field reversal (FFR), where electroosmosis should be absent. We found that during a series of measurements with monovalent salts, it was best to fill the cell starting from high ionic strength and decreasing the ionic strength during the series. The measurements with divalent salts were not sensitive to the filling procedure.

Polarization of a Diffuse Soft Particle Subjected to an Alternating Current Field

The polarization of a diffuse soft particle submerged in an aqueous electrolyte and subjected to a uniform alternating electric field is theoretically analyzed with the standard electrokinetic model (the Poisson-Nernst-Planck equations). The particle consists of a rigid uncharged core and a charged diffuse polyelectrolytic shell (soft layer) permeable to ions and solvent. Our focus is on the impact of the characteristics of the soft layer including the Donnan potential, the soft layer thickness, and the friction coefficient of the soft layer on the dipole coefficient, characterizing the strength of the polarization. Under the limits of thin double layers and thin polyelectrolytic shells, approximate analytical expressions to evaluate the dipole moment coefficients are derived for high-frequency and low-frequency ranges, respectively. The analytical results are compared and agree favorably with those numerically computed by the standard model. Interestingly, we discover that when the double layer is comparable to the soft layer the dipole moment behaves qualitatively differently at different Donnan potentials. When the Donnan potential is small, the dipole moment decreases as the double layer increases. In contrast, at large Donnan potentials, the dipole moment increases with the increase in the double layer. The distinct responses to Donnan potentials are attributed to the impact of the associated double layer on the charge distribution of mobile ions inside the soft layer. The theoretical model provides a fundamental basis for interpreting the polarization of heterogeneous systems, including environmental or biological colloids or microgel particles.

Equilibrium properties of charged spherical colloidal particles suspended in aqueous electrolytes: Finite ion size and effective ion permittivity effects

The equilibrium properties of a charged spherical colloidal particle immersed in an aqueous electrolyte solution are examined using an extension of the Standard Electrokinetic Model that takes into account the finite ion size by modeling the aqueous electrolyte solution as a suspension of polarizable insulating spheres in water. We find that this model greatly amplifies the steric effects predicted by the usual modified Poisson–Boltzmann equation, which only imposes a restriction on the ability of ions to approach one another. This suggests that a solution of the presented model under nonequilibrium conditions could have important consequences in the interpretation of dielectric and electrokinetic data in colloidal suspensions.

Directed Self-Assembly of Colloidal Crystals by Dielectrophoretic Ordering

In this Article, we report the dielectrophoretic assembly of colloidal particles and show how the kinetics of assembly and degree of ordering depend on the particle size, charge, solution ionic strength, and field strength and frequency. A special dielectrophoresis (DEP) sample cell is constructed and validated to quantitatively measure directed self-assembly via sequential light scattering and optical microscopy measurements. Our results confirm the recently established scaling for the order-disorder transition and extend it to higher scaled frequencies. The limiting scaling of the order-disorder transition and particle electrophoretic mobility are correctly predicted by the standard electrokinetic model (SEKM). In particular, the order-disorder transition line is predicted from the particle properties using a recently proposed empirical scaling law and the SEKM over an order of magnitude in particle size.

Nanoparticle ζ -Potentials

For over half a century, alternating electric fields have been used to induce particle transport, furnishing the ζ-potential of analytes with sizes ranging from a few nanometers to several micrometers. Concurrent advances in nanotechnology have provided new materials for catalysis, self-assembly, and biomedical applications, all of which benefit from a thorough understanding of particle surface charge. Therefore, the measurement of the ζ-potential via electrophoretic light scattering (ELS) has become essential for nanoparticle (NP) research. However, the interpretation of NP electrophoretic mobility, especially that of ligand-coated NPs, can be a complex undertaking. Despite the inherent intricacy of these data, key concepts from colloidal science can help to distill valuable information from ELS. In this Account, we adopt PEGylated Au NPs as an illustrative example to explore extensions of the classical theories of Smoluchowski, Hückel, and Henry to more contemporary theories for ligand-coated NP systems such as those from Ohshima, and Hill, Saville, and Russel. First, we review the basic experimental considerations necessary to understand NP electrophoretic mobility, identifying when O'Brien and White's numerical solution of the standard electrokinetic model should be adopted over Henry's closed-form analytical approximation. Next, we explore recent developments in the theory of ligand-coated particle electrophoresis, and how one can furnish accurate and meaningful relationships between measured NP mobility, ζ-potential, and surface charge. By identifying key ligand-coated NP parameters (e.g., coating thickness, permeability, molecular mass, and hydrodynamic segment size), we present a systematic method for quantitatively interpreting NP electrophoretic mobility. In addition to reviewing theoretical foundations, we describe our recent results that examine how the unique surface curvature of NPs alters and controls their properties. These data provide guidelines that can expedite the rational design of NPs for advanced uses, such as heterogeneous catalysis and in vivo drug delivery. As a practical demonstration of these concepts, we apply the ligand-coated theory to a recently developed noncovalent PEGylated Au NP drug-delivery system. Our analysis suggests that anion adsorption on the Au NP core may enhance the stability of these NP-drug conjugates in solution. In addition to providing useful nanochemistry insights, the information in this Account will be useful to biomedical and materials engineers, who use ELS and ζ-potentials for understanding NP dynamics.

Dielectric Relaxations of Ionic Thiol-Coated Noble Metal Nanoparticles in Aqueous Solutions: Electrical Characterization of the Interface

The radiowave dielectric properties of organothiol monolayer-protected Au and Ag metallic nanoparticles have been investigated in the frequency range of 10 kHz to 2 GHz, where a dielectric relaxation, due to the polarization of the ionic atmosphere at the aqueous interface, occurs. The simultaneous measurement of the particle size, by means of dynamic light scattering technique, and of the particle electrical charge, by means of laser microelectrophoresis technique, allow us to describe the whole dielectric behavior at the light of the standard electrokinetic model for charged colloidal particles. Au and Ag metallic nanoparticles experience a large charge renormalization, in agreement with the counterion condensation effect for charged spherical colloidal particles. The value of the effective valence Z(eff) of each nanoparticle investigated has been evaluated thanks to the dielectric parameters of the observed relaxation process and further confirmed by direct current electrical conductivity measurements. All in all, these results provide support for the characterization of the electrical interfacial properties of metallic nanoparticles by means of dielectric relaxation measurements.

Dielectric Properties of Micellar Aggregates Due to the Self-Assembly of Thermoresponsive Diblock Copolymers

The radiowave dielectric properties of aqueous solutions of thermosensitive copolymers, consisting of poly(2-acrylamido-2-methylpropanesulfonate) [PAMPS] and poly(N-isopropylacrylamide) [PNIPAAM] with different block lengths, have been investigated over a broad temperature and frequency range. These copolymers PAMPS(n)-b-PNIPAAM(m) form temperature responsive aggregates (micelles) that represent a class of self-assembled structures in water of great interest because of their potential use as drug delivery formulations and in diverse biotechnological applications. Copolymers formed by hydrophilic segments covalently attached to a hydrophobic segments are capable of forming a micellar structure as soon as the temperature is raised above their lower critical solution temperature. We have investigated the dielectric properties of PAMPS(n)-b-PNIPAAM(m) diblock copolymers with different lengths of the hydrophilic and hydrophobic segments during the whole aggregation process driven by the progressive increase of temperature. The process has been followed by the changes resulting in the dielectric parameters (the dielectric increment Δε and the relaxation frequency ν(0)) of the whole aqueous solution. The dielectric response of the micelles has been described within the framework of the standard electrokinetic model for charged colloidal particles, and the main characteristic parameters have been evaluated. Subsequent cross-linking of these diblock copolymers by a cationic PEO(x)-b-PAMPTMA(y) polyelectrolyte yields hybrid core-shell-corona systems, with the PNIPAAM hydrophobic blocks collapsed in the core, an interpolyelectrolyte chain complex forming the shell, and the hydrophilic PEO chains as an external corona. In this case too, the dielectric spectra can be appropriately accounted for within the same theoretical framework.

Electrokinetics of charged spherical colloidal particles taking into account the effect of ion size constraints

The electrokinetic properties of suspended spherical particles are examined using a modified standard electrokinetic model, which takes into account the finite ion size and considers that the minimum approach distance of ions to the particle surface need not be equal to their effective radius in the bulk solution. We calculate the conductivity increment and the electrophoretic mobility and present a detailed interpretation of the obtained results, based on the analysis of the equilibrium and field-induced ion concentrations, as well as the convective fluid flow in the neighborhood of the particle surface. We show that when charge reversal takes place, the sign of the concentration polarization remains unchanged while the sign of the electrophoretic mobility only changes under favorable circumstances.