Dynamic electrophoretic mobility of concentrated dispersions of spherical colloidal particles. On the consistent use of the cell model

Overcoming power-efficiency tradeoff in a micro heat engine by engineered system-bath interactions

All real heat engines, be it conventional macro engines or colloidal and atomic micro engines, inevitably tradeoff efficiency in their pursuit to maximize power. This basic postulate of finite-time thermodynamics has been the bane of all engine design for over two centuries and all optimal protocols implemented hitherto could at best minimize only the loss in the efficiency. The absence of a protocol that allows engines to overcome this limitation has prompted theoretical studies to suggest universality of the postulate in both passive and active engines. Here, we experimentally overcome the power-efficiency tradeoff in a colloidal Stirling engine by selectively reducing relaxation times over only the isochoric processes using system bath interactions generated by electrophoretic noise. Our approach opens a window of cycle times where the tradeoff is reversed and enables the engine to surpass even their quasistatic efficiency. Our strategies finally cut loose engine design from fundamental restrictions and pave way for the development of more efficient and powerful engines and devices.

Dynamic Electrophoresis of a Hydrophobic and Dielectric Fluid Droplet

Dynamic electrophoresis is the foundation for electroacoustical measurements, in which the electroacoustical signals may be used to analyze the size and electrostatic charge of colloidal entities by means of the results for dynamic electrophoretic mobility. Thus, the electrophoresis under an alternating electric field is the key foundation for electroacoustic theory. In this article, we develop a tractable analytical theory for the dynamic electrophoresis of hydrophobic and dielectric fluid droplets possessing uniform surface charge density. The tiny fluid droplets possess charged mobile surfaces and have found widespread applications in our day-to-day life. For dielectric fluid droplets (e.g., oil-water emulsions), the tangential electric stress at the interface is nonzero, which significantly affects its electrohydrodynamics under an oscillatory electric field, which has, however, a negligible impact on the electrophoretic motion of conducting droplets (e.g., mercury droplets). Besides, the micro/nanoscale fluid droplets often show hydrophobicity when they are immersed in an aqueous medium, and the impact of the electric field on hydrophobic surfaces remains a research frontier in the chemical discipline. Whereas a number of approximate expressions for electrophoretic mobility have been derived for the conducting droplet, none of them are applicable to such generic hydrophobic fluid droplets with dielectric permittivity that is significantly lower than or comparable to that of an aqueous medium. In this work, within the Debye-Hückel electrostatic framework, we elaborate an original analytical expression of dynamic electrophoretic mobility for this generic dielectric fluid droplet with a hydrophobic surface considering that the droplet retains its spherical shape during its oscillatory motion. We further derived a set of simplified expressions for dynamic electrophoretic mobility deduced under several limiting cases. The results are further illustrated, indicating the impact of pertinent parameters.

AC Electrokinetics of Salt-Free Multilayered Polymer-Grafted Particles

Interest in the electrical properties of the interface between soft (or polymer-grafted) nanoparticles and solutions is considerable. Of particular significance is the case of polyelectrolyte-coated particles, mainly taking into account that the layer-by-layer procedure allows the control of the thickness and permeability of the layer, and the overall charge of the coated particle. Like in simpler systems, electrokinetic determinations in AC fields (including dielectric dispersion in the 1 kHz-1 MHz frequency range and dynamic electrophoresis by electroacoustic methods in the 1-18 MHz range) provide a large amount of information about the physics of the interface. Different models have dealt with the electrokinetics of particles coated by a single polymer layer, but studies regarding multi-layered particles are far scarcer. This is even more significant in the case of so-called salt-free systems; ideally, the only charges existing in this case consist of the charge in the layer(s) and the core particle itself, and their corresponding countercharges, with no other ions added. The aims of this paper are as follows: (i) the elaboration of a model for the evaluation of the electrokinetics of multi-grafted polymer particles in the presence of alternating electric fields, in dispersion media where no salts are added; (ii) to carry out an experimental evaluation of the frequency dependence of the dynamic (or AC) electrophoretic mobility and the dielectric permittivity of suspensions of polystyrene latex spherical particles coated with successive layers of cationic, anionic, and neutral polymers; and (iii) finally, to perform a comparison between predictions and experimental results, so that it can be demonstrated that the electrokinetic analysis is a useful tool for the in situ characterization of multilayered particles.

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.

AC Conductivity Measurements of Ultradilute Colloidal Suspensions in HEPES Buffer

Impedance spectroscopy was used to probe the AC conductivity of extremely dilute colloidal suspensions (2.5 × 10^-5 ≤ Φ_w/v ≤ 4.0 × 10^-2) comprising of polystyrene microspheres (PS; κa ≫ 1 and ζ = -65 mV), gold nanoparticles (Au NPs; κa > 1 and ζ = -26 mV), and Au-coated PS metallodielectric particles (Au-PS) in HEPES buffer. When AC electric fields of strength 10 mV and 1 MHz were applied via 100 μm gap interdigitated microelectrodes across 10 μL samples, a highly resistive (θ_capacitive < 1°) and nonmonotonic response was obtained with particle concentrations at steady state. While the suspensions were less resistive (than the buffer) below a critical concentration, they became more resistive above it. More interestingly, particle-particle interactions took place in suspensions with concentrations as low as 0.005% w/v. We believe this unique behavior is linked to the ion size asymmetry in the HEPES molecule that provides an ideal microenvironment for counterionic polarization around the particles. The exact mechanism of polarization in HEPES, however, still remains elusive as the current theoretical models for simple electrolytes fail to explain our data.

Charging Poly(methyl Methacrylate) Latexes in Nonpolar Solvents: Effect of Particle Concentration

The electrophoresis of a well-established model system of charged colloids in nonpolar solvents has been studied as a function of particle volume fraction at constant surfactant concentration. Dispersions of poly(12-hydroxystearic acid)-stabilized poly(methyl methacrylate) (PMMA) latexes in dodecane were prepared with added Aerosol OT surfactant as the charging agent. The electrophoretic mobility (μ) of the PMMA latexes is found to decrease with particle concentration. The particles are charged by a small molecule charging agent (AOT) at finite concentration, and this makes the origin of this decrease in μ unclear. There are two suggested explanations. The decrease could either be due to the reservoir of available surfactant being exhausted at high particle concentrations or the interactions between the charged particles at high particle number concentrations. Contrast-variation small-angle neutron scattering measurements of PMMA latexes and deuterated AOT-d₃₄ surfactant in latex core contrast-matched solvent were used to study the former, and electrokinetic modeling was used to study the latter. As the same amount of AOT-d₃₄ is found to be incorporated with the latexes at all volume fractions, the solvodynamic and electrical interactions between particles are determined to be the explanation for the decrease in mobility. These measurements show that, for small latexes, there are interactions between the charged particles at all accessible particle volume fractions and that it is necessary to account for this to accurately determine the electrokinetic ζ potential.

Dynamic electrophoretic mobility and electric permittivity of concentrated suspensions of plate-like gibbsite particles

In this paper we present experimental results on the electrokinetic behavior of planar gibbsite particles in concentrated suspensions. The dc electrophoretic mobility measurements are in this case of little significance, as they are scarcely informative. In the present investigation, we show that the dielectric dispersion and dynamic electrophoresis can in contrast provide such information. The complicating factors are of course the non-spherical shape and the finite particle concentration, as no complete theory of these phenomena exists for such systems. We propose to use first of all a model of dynamic electrophoresis of spheroids in which the effect of volume fraction is considered by means of an approximate theory previously obtained for spheres, based on the evaluation of electrical and hydrodynamic interactions between particles. In addition, the role of volume fraction on the high frequency inertial relaxation is also ascertained and used to obtain a volume fraction-independent radius of the gibbsite spheroids. A similar approach is used for the evaluation of dielectric dispersion data. Both the dynamic mobility and dielectric constant dependencies on frequency were obtained for gibbsite suspensions of different volume fractions in 0.5mMKCl. The theoretical treatments elaborated were applied to these data, and a coherent picture of the geometrical and electrical characteristics of the particles was obtained.

Experimental and theoretical study of the silica particle interactions in the presence of multivalent rod-like ions

The silica particle interactions in the presence of spermidine were systematically investigated both from experimental and theoretical points of view. The hydrodynamic radii and the corresponding polydispersity indices of the colloidal silica particles were determined by dynamic light scattering (DLS) as a function of spermidine concentration. Whereas the effective size of the silica particles increases with increasing spermidine concentration (pointing to the particle aggregation), the polydispersity index first increases reaches a maximum and then further decreases with the increasing spermidine concentration. From the mobility measurements it was concluded that the increase in spermidine concentration causes less negative values of zeta potential, meaning that the adsorption of spermidine leads to the less negative silica surface. Moreover, Monte Carlo (MC) simulations also confirmed that the addition of spermidine reduces the repulsion between silica particles. The MC concentration profiles of spermidine close to the charged silica particle are in a very good agreement with the results obtained by theory. An important motivation for our study is the effectiveness of multivalent ions to coagulate colloidal suspensions; e.g., the multivalent ions are exploited in the water purification process.

Diverging electrophoretic and dynamic mobility of model silica colloids at low ionic strength in ethanol

Electroacoustics and laser Doppler electrophoresis were employed to measure the mobility of surface-modified silica colloids in ethanol as a function of the ionic strength. Sufficiently low volume fractions were chosen to exclude effects of interparticle interactions. At high ionic strength, the electrophoretic mobility μ(e) is equal to the (electroacoustic) dynamic mobility μ(d) at 3.3 MHz. However, the ratio μ(d)/μ(e) increases significantly to ∼5 at low ionic strength. This increase may be related to the porous outer layer of the surface-modified silica spheres.

Electrodynamics of soft multilayered particles dispersions: dielectric permittivity and dynamic mobility

A theory is presented for the electrodynamics of dispersions of spherical soft multilayered (bio)particles consisting of a hard core surrounded by step-function or diffuse-like polymeric layers with distinct electrohydrodynamic and structural features.

Effects of non-equilibrium association-dissociation processes in the dynamic electrophoretic mobility and dielectric response of realistic salt-free concentrated suspensions

Electrokinetic investigations in nanoparticle suspensions in aqueous media are most often performed assuming that the liquid medium is a strong electrolyte solution with specified concentration. The role of the ions produced by the process of charging the surfaces of the particles is often neglected or, at most, the concentrations of such ions are estimated in some way and added to the concentrations of the ions in the externally prepared solution. The situation here considered is quite different: no electrolyte is dissolved in the medium, and ideally only the counterions stemming from the particle charging are assumed to be in solution. This is the case of so-called salt-free systems. With the aim of making a model for such kind of dispersions as close to real situations as possible, it was previously found to consider the unavoidable presence of H(+) and OH(-) coming from water dissociation, as well as the (almost unavoidable) ions stemming from the dissolution of atmospheric CO2. In this work, we extend such approach by considering that the chemical reactions involved in dissociation and recombination of the (weak) electrolytes in solution must not necessarily be in equilibrium conditions (equal rates of forward and backward reactions). To that aim, we calculate the frequency spectra of the electric permittivity and dynamic electrophoretic mobility of salt-free suspensions considering realistic non-equilibrium conditions, using literature values for the rate constants of the reactions. Four species are linked by such reactions, namely H(+) (from water, from the--assumed acidic--groups on the particle surfaces, and from CO2 dissolution), OH(-) (from water), HCO3(-) and H2CO3 (again from CO2). A cell model is used for the calculations, which are extended to arbitrary values of the surface charge, the particle size, and particle volume fraction, in a wide range of the field frequency ω. Both approaches predict a high frequency relaxation of the counterion condensated layer and a Maxwell-Wagner-O'Konski electric double layer relaxation at intermediate frequencies. Also, in both cases an inertial decay of the electrophoretic mobility at high ω takes place. The most significant difference between the present model and previous results based on the equilibrium hypothesis is by no means negligible: only in non-equilibrium conditions do we find a low-frequency relaxation (mostly noticed in permittivity data, while its significance is lower in dynamic mobility spectra). This new relaxation presents all the characteristic features of the concentration polarization (or alpha) dispersion. These are: i) the average electric polarization of the system increases when the relaxation frequency is surpassed, contrary to the behavior after Maxwell-Wagner type relaxations; ii) the amplitude of the relaxation increases with surface charge, reaching a sort of saturation if the charge is too high; iii) the relaxation frequency increases with volume fraction while the relaxation amplitude decreases; iv) the characteristic frequency is reduced by the increase in particle radius. All these facts confirm that the non-equilibrium approach seems to better describe the physics of the system by giving rise to a concentration polarization kind of relaxation, only possible when ions can accumulate on both sides of the particles as dictated by the field, and not as determined by equilibrium conditions in the dissociation-recombination reactions involved.

Electrophoresis and dielectric dispersion of spherical polyelectrolyte brushes

Spherical polyelectrolyte brushes (SPBs) consist of a rigid core on which polyelectrolyte chains are grafted in such a way that in certain conditions (low ionic strength and high charge of the chains) the polymer chains extend radially toward the liquid medium. Because of the hairy-like structure of the polymer brushes, the typical soft-particle approach used for explaining the behavior of polyelectrolyte-coated particles must be modified, using the assumptions that the density of charged segments in the polymer chains decreases with the squared distance to the rigid core surface and that the same happens to the friction between the brushes and the surrounding fluid. Interest in clarifying the electrokinetics of these systems is not just academic. It has recently been found experimentally (Jiménez et al., Soft Matter 2011, 7, 3758-3762) that the response of concentrated suspensions of spherical polyelectrolyte brushes in the presence of alternating electric fields shows a number of unexpected features. Both dielectric and dynamic electrophoretic mobility spectra (respectively, dependences of the electric permittivity and the AC electrophoretic mobility on the frequency of the applied field) showed very special aspects, with giant values of the mobility and an unusually strong dielectric relaxation in the kHz region. In the present paper we give a full account of the electrodynamics of such systems, based on a cell model for describing the hydrodynamic and electrical interactions between the particles. It is found that the low-frequency dynamic mobility of SPBs is much higher than that of rigid particles of comparable size and charge, making any interpretation based on zeta potential estimations of very limited applicability. The very characteristic feature of SPBs in concentrated suspensions, namely, the enhanced alpha relaxation, can be explained by considering an adequate description of the field-induced perturbations in the counterion and co-ion concentrations, well developed both outside and inside the soft layer in the case of brush-coated particles. It can be also pointed out that the dynamic electrophoretic mobility of SPBs increases with the volume fraction of particles, as a consequence of the large thickness of the brush. Predictions are also shown for the effects of friction coefficient and charge of the polyelectrolyte layer. The results compare well with experimental spectra of the dynamic mobility and electric permittivity of moderately concentrated suspensions of SPBs consisting of a 50 nm polystyrene core with grafted poly(styrene sulfonate) chains some 140 nm in length.

Adsorption of anionic and cationic surfactants on anionic colloids: supercharging and destabilization

We present herein a study on the adsorption of anionic (SDS), cationic (CTAB), and nonionic (C(12)E(5)) surfactants onto anionic silica nanoparticles. The effects of this adsorption are studied by means of the static structure factor, S(q), and the collective diffusion coefficient, D(c), obtained from small-angle X-ray scattering and dynamic light scattering measurements, respectively. The effective charge on the particles was determined also from classical electrophoresis and electroacoustic sonic-amplitude measurements. The surface tension of the sample was also investigated. Of particular note is the adsorption of SDS onto the silica nanoparticles, which leads to supercharging of the interface. This has interesting repercussions for structures obtained by the layer-by-layer (LbL) technique, because emulsions stabilized with supercharged and hydrophobized silica are perfect candidates for use in a multilayer system.

The actual dielectric response function for a colloidal suspension of spherical particles

In this paper, we present a theoretical analysis of the dielectric response of a dense suspension of spherical colloidal particles based on a self-consistent cell model. Particular attention is paid to (a) the relationship between the dielectric response and the conductivity response and (b) the connection between the real and imaginary parts of these responses based on the Kramers-Kronig relations. We have thus clarified the analysis of Carrique et al. (Carrique, F.; Criado, C.; Delgado, A. V. J. Colloid Interface Sci. 1993, 156, 117). We have shown that both the conduction and displacement current components are complex quantities with both real and imaginary parts being frequency dependent. The dielectric response exhibits characteristics of two relaxation phenomena: the Maxwell-Wagner and the alpha-relaxations, with the imaginary part being the more sensitive instrument. The inverse Fourier transform of the simulated dielectric response is compared with a phenomenological, two-exponential response function with good agreement obtained. The two fitted decay times also compare well with times extracted from the explicit simulations.

High-frequency behavior of the dynamic mobility and dielectric response of concentrated colloidal dispersions

A matched asymptotic analysis of the system of equations governing the electrokinetic cell model of ref 4 (Ahualli, S.; Delgado, A.; Miklavcic, S.; White, L. R. Langmuir 2006, 22, 7041) is performed. Asymptotic expressions are obtained for the dynamic mobility and complex conductivity response of a dense suspension of charged spherical particles to an applied electric field. The asymptotic expressions are compared with full numerical calculations of the linear response functions as a function of surface (zeta) potential, electrolyte strength, and particle density. We find that the numerical procedure used is robust and highly accurate at a very high frequency under a wide range of double-layer conditions. The asymptotic form for the dielectric response of the system is accurate to megahertz frequencies. The asymptotic formulas for the other response functions have limited viability as predictive tools within the current range of experimentally accessible frequencies but are useful as checks on numerical calculations.

Role of surface conductivity in the dynamic mobility of concentrated suspensions

In this article, a cell model is used for the evaluation of the alternating current (ac) mobility (dynamic mobility) of spherical particles in suspensions of arbitrary volume fractions of solids. The main subject is the consideration of the role of the electrical conductivity (SLC or K(sigmai)) of the stagnant layer (SL) on the mobility. It is assumed that the total surface conductivity (K(sigma)), resulting from both K(sigmai) and the diffuse layer conductivity (K(sigmad)), is constant in the cases considered and that it is the K(sigmai)-K(sigmad) balance that determines the SL effects. We first explore the effect of K(sigmai) on the frequency dependence of the dynamic mobility. It is found that the mobility decreases on average, for any frequency, when K(sigmai) increases. This is a consequence of stagnancy: ions in the SL, although contributing to the surface conductivity, do not drag liquid with them when they migrate and do not contribute to electro-osmotic flow or, equivalently, to electrophoresis. Three relaxations are observed in the mobility-frequency spectrum: inertial (the particle and liquid motions are hindered), Maxwell-Wagner-O'Konski (ions in the double layer cannot follow the field oscillations and can move only over a distance much smaller that the diffuse layer thickness), and the so-called alpha or concentration polarization process (the ions can rearrange around the particle, but they cannot form the electrolyte concentration field that appears at low frequency). Whereas the first two relaxations are little affected by K(sigmai), the alpha process undergoes significant changes. Thus, the mobility increases with frequency around the alpha relaxation region if K(sigmai) is negligible, but it decreases with frequency in the same interval if K(sigmai) is finite. With the aim of explaining this behavior, we calculate the capillary osmosis velocity field that is the fluid flow provoked by the concentration gradient around the particle. The calculations presented demonstrate that the velocity is reduced (for each frequency and position) when the SLC is raised. It is proposed that such a decrease adds to that due to the changes in the induced dipole moment of the particle, also favoring a decrease in the mobility. These tendencies are also present when the volume fraction of solids, phi, is modified, although higher phi values somewhat hide the effect of K(sigmai), as in fact observed with all features of electrokinetics associated with the phenomenon of concentration polarization.

Dynamic mobility of rodlike goethite particles

In this work we consider how the spheroidal shape of colloidal particles and their concentration in suspension influence their electrokinetic properties in alternating (ac) electric fields, in particular, their electrophoretic mobility, traditionally known as dynamic mobility in the case of ac fields. Elaboration of a formula for the mobility is based on two previous models related to the electrokinetic response of spheroids in dilute suspensions, completed by means of an approximate formula to account for the finite concentration of particles. At the end, semianalytical formulas have been obtained in the form of the classical Helmholtz-Smoluchowski equation for the mobility with three frequency-dependent factors, each dealing with inertia relaxation, electric double layer polarization and volume fraction effects. The two resulting expressions differ basically in their consideration of double layer polarization processes, as one considers only Maxwell-Wagner-O'Konski polarization (related to the mismatch between the conductivities of the particles plus their double layers and the liquid medium), and the other also includes the concentration polarization effect. Since in the frequency range typically used in dynamic mobility measurements the latter polarization has already relaxed, both models are capable of accounting for the dynamic mobility data experimentally obtained on elongated goethite particles in the 1-18 MHz frequency range. Results are presented concerning the effects of volume fraction, ionic strength, and pH, and they indicate that the models are good descriptions of the electrokinetics of these systems, and that dynamic mobility is very sensitive not only to the zeta potential of the particles, but also to their concentration, shape, and average size, and to the stability of the suspensions. The effects of ionic strength and pH on the dynamic mobility are very well captured by both models, and a consistent description of the dimensions and zeta potentials of the particles is reached. Increasing the volume fraction of the suspensions produces mobility variations that are only partially described by the theoretical calculations due to the likely flocculation of the particles, mainly associated with the fact that goethite particles are not homogeneously charged, with attraction between positive and negative patches being possible.

Dynamic dielectric response of concentrated colloidal dispersions: comparison between theory and experiment

The cell-model electrokinetic theory of Ahualli et al. Langmuir 2006, 22, 7041; Ahualli et al. J. Colloid Interface Sci. 2007, 309, 342; and Bradshaw-Hajek et al. Langmuir 2008, 24, 4512 is applied to a dense suspension of charged spherical particles, to exhibit the system's dielectric response to an applied electric field as a function of solids volume fraction. The model's predictions of effective permittivity and complex conductivity are favorably compared with published theoretical calculations and experimental measurements on dense colloidal systems. Physical factors governing the volume fraction dependence of the dielectric response are discussed.

AC electrokinetics of concentrated suspensions of soft particles

In this work, we show how the cell model traditionally used for the evaluation of the electrokinetic properties of concentrated suspensions can be modified to include the case of soft particles, that is, particles consisting of a rigid core and a polyelectrolyte membrane. The Navier-Stokes and Poisson's equations have been modified to account for the presence of extra friction and a volume-distributed charge in the membrane. In addition to the boundary conditions on the particle and the cell boundary, it is necessary to define conditions on the polymer-electrolyte solution interface. The frequency dependence of the dynamic mobility and electric permittivity of suspensions of soft particles with arbitrary solids concentration is computed. It is shown that the dynamic mobility of these systems is larger than that corresponding to hard particles with the same charge. For the permittivity, the same trends are observed: the R-relaxation amplitude increases upon coating. It is found that friction plays an important role in determining the mobility, while the permittivity is more affected by the concentration of solids. The model also predicts that the charges on the core and in the membrane are very important parameters, although their effects differ on the mobility and the permittivity. While the former depends mainly on the membrane charge, the latter is responsive to both charges at comparable extents.

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.

Frequency-dependent electrical conductivity of concentrated dispersions of spherical colloidal particles

This paper outlines the application of a self-consistent cell-model theory of electrokinetics to the problem of determining the electrical conductivity of a dense suspension of spherical colloidal particles. Numerical solutions of the standard electrokinetic equations, subject to self-consistent boundary conditions, are implemented in formulas for the electrical conductivity appropriate to the particle-averaged cell model of the suspension. Results of calculations as a function of frequency, zeta potential, volume fraction, and electrolyte composition, are presented and discussed.

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.

Use of a cell model for the evaluation of the dynamic mobility of spherical silica suspensions

In this paper we evaluate the validity of a cell model for the calculation of the dynamic mobility of concentrated suspensions of spheres. The key point is the consideration of the boundary conditions (electrical and hydrodynamic) at the boundary of the fluid cell surrounding a single probe particle. The model proposed is based on a universal criterion for the averages of fluid velocity, electric potential, pressure field or electrochemical properties in the cell. The calculations are checked against a wide set of experimental data on the dynamic mobility of silica suspensions with two different radii, several ionic strengths, and two particle concentrations. The comparison reveals an excellent agreement between theory and experiment, and the model appears to be extremely suitable for correctly predicting the behavior of the dynamic mobility, including the changes in the zeta potential, zeta, with ionic strength, the frequency and amplitude of the Maxwell-Wagner-O'Konski relaxation, and the inertial relaxation occurring at the top of the frequency range accessible to our experimental device.

Cell Models for the Primary Electroviscous Effect

The primary electroviscous effect in a nondilute suspension of charged spherical particles is studied by means of cell models. The governing equations are derived, and then analytic results are obtained by restricting attention to the limit of thin double layers, small Hartmann and Peclet numbers, and small potentials. Previous work has assumed that the velocity at the outer boundary of the cell is identical to the imposed flow, as proposed by Simha (J. Appl. Phys. 1952, 23, 1020). Results with this boundary condition are compared against those predicted when the tangential shear stress on the outer boundary is assumed to be unperturbed, as proposed by Happel (J. Appl. Phys. 1957, 28, 1288). Both the hydrodynamic and electroviscous contributions to the effective viscosity are smaller with the Happel boundary condition, showing that such cell models offer a range of predictions and should be used with caution.