[Show abstract][Hide abstract] ABSTRACT: 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.
[Show abstract][Hide abstract] ABSTRACT: A long-lasting experience in the electrokinetics of suspensions has shown that the so-called standard model may be partly in error in explaining experimental data. In this model, the stagnant layer is considered nonconducting (Ksigmai=0), and only the diffuse layer contributes to the total surface conductivity (Ksigma=Ksigmad). In the present work, the authors analyze the consequences of assuming a nonzero stagnant layer conductivity on the permittivity of concentrated suspensions. Using a cell model to account for the particle-particle interactions, and a well established ion adsorption isotherm on the inner region of the double layer, the authors find the frequency-dependent electric permittivity of suspensions of spherical particles with volume fractions of solids up to above 40%. It is demonstrated that the addition of Ksigmai significantly increases the contributions of the double layer to the polarization of the suspension: the alpha or concentration polarization at low (kilohertz) frequencies, and the Maxwell-Wagner-O'Konski (associated with conductivity mismatch between particle and medium) one at intermediate (megahertz) frequencies. While checking for the possibility that the results obtained in conditions of Ksigmai not equal 0 could be reproduced assuming Ksigmai=0 and raising Ksigmad to reach identical total Ksigma, it is found that this is approximately possible in the calculation of the permittivity. Interestingly, this does not occur in the case of electrophoretic mobility, where the situations Ksigma=Ksigmad and Ksigma=Ksigmad+Ksigmai (for equal Ksigma) can be distinguished for all frequencies. This points to the importance of using more than one electrokinetic technique to properly evaluate not only the zeta potential but other transport properties of concentrated suspensions, particularly Ksigmai.
The Journal of Chemical Physics 04/2007; 126(10):104903. DOI:10.1063/1.2538679 · 2.95 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In the past few years, different models and analytical approximations have been developed facing the problem of the electrical conductivity of a concentrated colloidal suspension, according to the cell-model concept. Most of them make use of the Kuwabara cell model to account for hydrodynamic particle-particle interactions, but they differ in the choice of electrostatic boundary conditions at the outer surface of the cell. Most analytical and numerical studies have been developed using two different sets of boundary conditions of the Neumann or Dirichlet type for the electrical potential, ionic concentrations or electrochemical potentials at that outer surface. In this contribution, we study and compare numerical conductivity predictions with results obtained using different analytical formulas valid for arbitrary zeta potentials and thin double layers for each of the two common sets of boundary conditions referred to above. The conductivity will be analyzed as a function of particle volume fraction, phi, zeta potential, zeta, and electrokinetic radius, kappaa (kappa(-1) is the double layer thickness, and a is the radius of the particle). A comparison with some experimental conductivity results in the literature is also given. We demonstrate in this work that the two analytical conductivity formulas, which are mainly based on Neumann- and Dirichlet-type boundary conditions for the electrochemical potential, predict values of the conductivity very close to their corresponding numerical results for the same boundary conditions, whatever the suspension or solution parameters, under the assumption of thin double layers where these approximations are valid. Furthermore, both analytical conductivity equations fulfill the Maxwell limit for uncharged nonconductive spheres, which coincides with the limit kappaa --> infinity. However, some experimental data will show that the Neumann, either numerical or analytical, approach is unable to make predictions in agreement with experiments, unlike the Dirichlet approach which correctly predicts the experimental conductivity results. In consequence, a deeper study has been performed with numerical and analytical predictions based on Dirichlet-type boundary conditions.
The Journal of Physical Chemistry B 03/2006; 110(12):6179-89. DOI:10.1021/jp057030e · 3.30 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In the last few years, different theoretical models and analytical approximations have been developed addressing the problem of the electrical conductivity of a concentrated colloidal suspension. Most of them are based on the cell model concept, and coincide in using Kuwabara's hydrodynamic boundary conditions, but there are different possible approaches to the electrostatic boundary conditions. We will call them Levine-Neale's (LN, they are Neumann type, that is they specify the gradient of the electrical potential at the boundary), and Shilov-Zharkikh's (SZ, Dirichlet type). The important point in our paper is that we show by direct numerical calculation that both approaches lead to identical evaluations of the conductivity of the suspensions if each of them is associated to its corresponding evaluation of the macroscopic electric field. The same agreement between the two calculations is reached for the case of electrophoretic mobility. Interestingly, there is no way to reach such identity if two possible choices are considered for the boundary conditions imposed to the field-induced perturbations in ionic concentrations on the cell boundary (r = b), deltan(i) (r = b). It is demonstrated that the conditions deltan(i)(b) = 0 lead to consistently larger conductivities and mobilities. A qualitative explanation is offered to this fact, based on the plausibility of counter-ion diffusion fluxes favoring both the electrical conduction and the motion of the particles.
Advances in Colloid and Interface Science 01/2006; 118(1-3):43-50. DOI:10.1016/j.cis.2005.04.001 · 7.78 Impact Factor