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

Effects of ionic valence on aggregation kinetics of colloidal particles with and without a mixing flow

HYPOTHESIS

The classical Schulze-Hardy rule states that the critical coagulation concentration (CCC) of colloidal particles is inversely proportional to the counter-ionic valence at powers ranging from 2 to 6. However, the inverse Schulze-Hardy rule has recently been proposed, suggesting that the CCC can also be inversely proportional to the co-ionic valence. Previous studies on these rules did not consider the effect of flow on aggregation kinetics and the CCC. This study aims to investigate the effect of multivalent counter-ions and co-ions on aggregation kinetics and the CCCs in systems with and without a mixing flow.

EXPERIMENTS

We measured the aggregation rate coefficients of polystyrene sulfate latex particles as a function of the salt concentration with different ionic species. Furthermore, we analyzed these measurements using theoretical models based on hydrodynamic pair-diffusion in a random flow and trajectory analysis in two steady flows. The analysis was conducted using zeta potentials determined through electrophoretic measurements.

FINDINGS

Although the trajectory analysis underestimates the CCCs, the hydrodynamic pair-diffusion model can capture the shift of critical coagulation concentrations in the mixing flow to higher values than those in Brownian aggregation and also shows a better agreement with the experimental results. This result suggests that combining random flow and Brownian diffusion is crucial for developing a consistent framework for predicting both Brownian aggregation and aggregation in a mixing flow.

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.

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.

Convective and diffusive effects on particle transport in asymmetric periodic capillaries

We present here results of a theoretical investigation of particle transport in longitudinally asymmetric but axially symmetric capillaries, allowing for the influence of both diffusion and convection. In this study we have focused attention primarily on characterizing the influence of tube geometry and applied hydraulic pressure on the magnitude, direction and rate of transport of particles in axi-symmetric, saw-tooth shaped tubes. Three initial value problems are considered. The first involves the evolution of a fixed number of particles initially confined to a central wave-section. The second involves the evolution of the same initial state but including an ongoing production of particles in the central wave-section. The third involves the evolution of particles a fully laden tube. Based on a physical model of convective-diffusive transport, assuming an underlying oscillatory fluid velocity field that is unaffected by the presence of the particles, we find that transport rates and even net transport directions depend critically on the design specifics, such as tube geometry, flow rate, initial particle configuration and whether or not particles are continuously introduced. The second transient scenario is qualitatively independent of the details of how particles are generated. In the third scenario there is no net transport. As the study is fundamental in nature, our findings could engender greater understanding of practical systems.

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.

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 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.

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.