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

Electrokinetic and dielectric response of a concentrated salt-free colloid: Different approaches to counterion finite-size effects

In the present work, a general model is developed for the electrokinetics and dielectric response of a concentrated salt-free colloid that takes into account the finite size of the counterions released by the particles to the solution. The effects associated with the counterion finite size have been addressed using a hard-sphere model approach elaborated by Carnahan and Starling [N. F. Carnahan and K. E. Starling, Equation of state for nonattracting rigid spheres, J. Chem. Phys. 51, 635 (1969)0021-960610.1063/1.1672048]. A more simple description of the finite size of the counterions based on that by Bikerman has also been considered for comparison. The studies carried out in this work include predictions on the effect of the finite counterion size on the equilibrium properties of the colloid and its electrokinetic and dielectric response when it is subjected to constant or alternating electric fields. The results show how important the counterion finite-size effects are for most of the electrokinetic and dielectric properties of highly charged and concentrated colloids, mainly for the static and dynamic electrophoretic mobilities. Furthermore, new insights are provided on the counterion condensation effect when counterions are allowed to have finite size. Focus is placed on the changes undergone by their concentration in the condensation layer for low-salt and highly charged colloids.

Electrokinetic detection of the salt-free condition in colloids. Application to polystyrene latexes

Because of their singular phenomenology, the so-called salt-free colloids constitute a special family of dispersed systems. Their main characteristic is that the dispersion medium ideally contains only the solvent and the ions compensating exactly the surface charge of the particles. These ions (often called released counterions) come into the solution when the surface groups responsible for the particles charge get ionized. An increasing effort is nowadays dedicated to rigorously compare theoretical model predictions for ideal salt-free suspensions, where only the released counterions are supposed to be present in solution, with appropriately devised experiments dealing with colloids as close as possible to the ideal salt-free ones. Of course, if the supporting solution is aqueous, the presence of atmospheric contamination and any other charged species different from the released counterions in the solution must be avoided. Because this is not an easy task, the presence of dissolved atmospheric CO₂ and of H⁺ and OH^- from water dissociation cannot be fully discarded in aqueous salt-free solutions (often denominated realistic in such case). Ultimately, at some point, the role of the released counterions will be comparable or even larger in highly charged concentrated colloids than that of added salts. These topics are covered in the present contribution. The model results are compared with experimental data on the dynamic mobility and dielectric dispersion of polystyrene spheres of various charges and sizes. As a rule, it is found that the model correctly predicts the significance of alpha and Maxwell-Wagner-O'Konski relaxations. Positions and amplitudes of such relaxations are well predicted, although it is necessary to assume that the released counterions are potassium or sodium instead of protons, otherwise the frequency spectra of experimental mobility and permittivity differ very significantly from those theoretically calculated. The proposed electrokinetic evaluation is an ideal tool for detecting in situ the possible contamination (or incomplete ion exchange of the latexes). A satisfactory agreement is found when potassium counterions are assumed to be in solution, mostly if one considers that the comparison is carried out without using any adjustable parameters.

Transient electrophoresis of a conducting spherical particle embedded in an electrolyte-saturated Brinkman medium

In this study, the time-dependent electrophoretic motion of a conducting spherical particle embedded in an arbitrary electrolyte solution saturated porous medium is investigated. The porous medium is uniformly charged and the embedded hard particle is charged with constant ζ -potential or constant surface charge density. The unsteady modified Brinkman equation with an electric force term, which governs the fluid velocity field, is used to model the porous medium and is solved by Laplace's transform technique. An analytical expression for the electrophoretic velocity of the spherical particle is obtained in Laplace transform domain as a function of the relevant parameters, and its inversion is obtained through numerical techniques. Also, in this study, the steady-state electrophoretic velocity is obtained analytically as linear functions of  ζ -potential (or surface density charge) and the fixed charge density. The steady-state electrophoretic velocity is displayed graphically for various relevant parameters and compered with the available data in the literature. Also, the numerical values of the transient electrophoretic velocity are plotted versus the nondimensional elapsed time and discussed for different values of the Debye length parameter, density ratio, permeability of the porous medium, and for high and nonconducting particles.

Transient electrophoresis in a suspension of charged particles with arbitrary electric double layers

The startup of electrophoretic motion in a suspension of spherical colloidal particles, which may be charged with constant zeta potential or constant surface charge density, due to the sudden application of an electric field is analytically examined. The unsteady modified Stokes equation governing the fluid velocity field is solved with unit cell models. Explicit formulas for the transient electrophoretic velocity of the particle in a cell in the Laplace transforms are obtained as functions of relevant parameters. The transient electrophoretic mobility is a monotonic decreasing function of the particle-to-fluid density ratio and in general a decreasing function of the particle volume fraction, but it increases and decreases with a raise in the ratio of the particle radius to the Debye length for the particles with constant zeta potential and constant surface charge density, respectively. On the other hand, the relaxation time in the growth of the electrophoretic mobility increases substantially with an increase in the particle-to-fluid density ratio and with a decrease in the particle volume fraction but is not a sensitive function of the ratio of the particle radius to the Debye length. For specified values of the particle volume fraction and particle-to-fluid density ratio in a suspension, the relaxation times in the growth of the particle mobility in transient electrophoresis and transient sedimentation are equivalent.

Start-Up Electrophoresis of a Cylindrical Particle with Arbitrary Double Layer Thickness

The start-up of electrophoretic motion of a charged circular cylindrical particle in an unbounded solution of arbitrary electrolytes is analytically investigated. The modified Stokes equation for the transient fluid flow field is solved by using the Laplace transform. Analytical formulas for the time-evolving electrophoretic velocities of the dielectric cylinder are determined for the transversely and axially imposed electric fields, and they can be superimposed linearly for an imposed electric field of arbitrary direction. The transient electrophoretic velocities normalized by their respective steady-state values increase monotonically with an increase in the ratio of the particle radius to the Debye screening length but decrease monotonically with an increase in the particle-to-fluid density ratio, keeping the other parameter unchanged. The normalized electrophoretic acceleration of the particle decreases monotonically with the elapsed time. In general, the electrophoretic velocity of the cylindrical particle is not collinear with the arbitrarily oriented imposed electric field. The effect of the relaxation time for the transient electrophoresis is much more important for a cylindrical particle than for a spherical particle.

Influence of ion size effects on the electrokinetics of aqueous salt-free colloids in alternating electric fields

Electrokinetics is the science of the physical phenomena appearing at the solid-liquid interface of dispersed particles subjected to external fields. Techniques based on electrokinetic phenomena constitute an important set of tools for the electrical characterization of colloids because of their sensitivity to the properties of particle-solution interfaces. Their rigorous description may require inclusion of the effects of finite size of chemical species in the theoretical models, and, particularly in the case of salt-free (no external salt added) aqueous colloids, also consideration of water dissociation and possible carbon dioxide contamination in the aqueous solution. A new ac electrokinetic model is presented for concentrated salt-free spherical colloids for arbitrary characteristics of the particles and aqueous solution, including finite-size effects of chemical species by appropriate modifications of the chemical reaction equations to include such non-ideal aspects. The numerical solution of the electrokinetic equations in an alternating electric field has also been carried out by using a realistic non-equilibrium scenario accounting for association-dissociation processes in the chemical reactions. The results demonstrate the importance of including finite-size effects in the electrokinetic response of the colloid, mainly at high frequencies of the electric field, and for highly charged colloids. Findings of previous models for pointlike ions or for ideal salt-free colloids including finite ion size effects are recovered with the present model, for the appropriate limiting conditions.

Non-monotonic concentration dependence of the electro-phoretic mobility of charged spheres in realistic salt free suspensions

Using super-heterodyne Doppler velocimetry with multiple scattering correction, we extend the optically accessible range of concentrations in experiments on colloidal electro-kinetics. Here, we measured the electro-phoretic mobility and the DC conductivity of aqueous charged sphere suspensions covering about three orders of magnitude in particle concentrations and transmissions as low as 40%. The extended concentration range for the first time allows the demonstration of a non-monotonic concentration dependence of the mobility for a single particle species. Our observations reconcile previous experimental observations made on other species over restricted concentration ranges. We compare our results to the state-of-the-art theoretical calculations using a constant particle charge and the carefully determined experimental boundary conditions as input. In particular, we consider the so-called realistic salt free conditions, i.e., we respect the release of counterions by the particles, the solvent hydrolysis, and the formation of carbonic acid from dissolved neutral CO₂. We also compare our results to previous results obtained under similarly well-defined conditions. This allows identification of three distinct regions of differing density dependence. There is an ascent during the build-up of double layer overlap, which is not expected by theory, an extended plateau region in quantitative agreement with theoretical expectation based on a constant effective charge and a sudden decrease, which occurs way before the expected gradual decrease. Our observations suggest a relation of the non-monotonic behavior to a decrease in particle charge, and we tentatively discuss possibly underlying mechanisms.