Cell Models for the Primary Electroviscous Effect

Dynamic viscosity of colloidal silica suspensions at low and high volume fractions

A comprehensive study was carried out on the dynamic viscosity of X30 silica dispersions at both high and low volume fractions of colloidal silica particles at various electrolyte ionic strength and pH values. Booth and Ruiz-Reina and Carrique theoretical models (R-R&C) were compared in predicting the primary electroviscous effect (PEE) for viscosity at low volume fractions. To this respect the colloidal dispersion was well characterised with regards to electrolyte properties such as the Debye length, κ^-1, calculated from the ionic strength, and zeta potential, ζ, calculated from the electrophoretic mobility using the full numerical model by O'Brien and White (O'B&W). R-R&C hard sphere model (which is a modified version of Simha hard sphere model that includes a boundary condition by Happel on the outer radius of the cell) and the semi-empirical Krieger-Dougherty (K-D) models were fitted to the experimental data at high volume fractions. At both low and high volume fractions the viscosity increased with pH and decreased with ionic strength. At low volume fractions both theoretical models significantly underestimated the experimental dynamic viscosities obtained in this work. This could be attributed to the fuzzy structures for silica particles in aqueous conditions reported previously in the literature, where a significantly larger electroviscous parameter, p, was obtained experimentally for silica particles. The experimental electroviscous parameter, p_exp, in this work was found to be roughly an order of magnitude up to 26 times larger than that predicted by Booth and around 5 ± 1 times larger than the predicted p by R-R&C model allowing the introduction of a correction factor in the PEE coefficient obtained from R-R&C model enabling good prediction for X30 silica dispersions by the latter model. The significant improvement of the electroviscous effect predictions by R-R&C model compared to Booth model may be attributed to the limitations invalidating the Booth model at the electrolyte conditions in this study. At high volume fractions, the R-R&C hard sphere cell model, gave a much better fit to the experimental data compared to the K-D model, which also had the advantage of being only dependent on a coefficient that linearly relates an effective volume fraction postulated for the fuzzy silica particles to the experimental. The K-D model however, depends on the intrinsic viscosity, [η], which requires the calculation of the experimental slope of the dynamic viscosity against volume fraction in the dilute limit, and also on a maximum packing fraction as a fitting parameter. Due to the effect of the fuzzy structures on the viscosity, the latter effective volume fraction ϕ_eff was calculated using two approaches: (i) as a fitting parameter by fitting the R-R&C hard sphere model to the experimental viscosity data over the entire volume fraction range, and (ii) by fitting only the linear part of the experimental viscosity at low volume fractions, It is concluded that the R-R&C hard sphere model with the effective volume fraction accounting for the fuzzy structures fits reasonably well the full range of experimental results at low and high volume fractions. When the model was used with the Adamczyk effective volume fraction (i.e., considering only the dilute region in the fitting procedure), the predictions became worse at high volume fractions where significant deviations from the experimental results were found upon the increase of pH.

Effective viscosity of a concentrated suspension of uncharged spherical soft particles

We present a theory of the effective viscosity eta(s) of a concentrated suspension of uncharged polymer-coated spherical particles, which are termed uncharged spherical soft particles, in a liquid of viscosity eta on the basis of a cell model. These particles consist of the uncharged particle core of radius a covered with an uncharged polymer layer of thickness d (a polymer-coated particle thus has an inner radius a and an outer radius b = a + d). We assume that polymer segments are regarded as resistance centers, exerting frictional forces--gamma u on the liquid flowing in the polymer layer, where u is the liquid velocity and gamma is the frictional coefficient. We derive an analytic expression for the effective viscosity eta(s) of the suspension, which depends on the radii a and b, and volume fraction phi of the spheres and a parameter lambda = (gamma/eta)(1/2). The obtained expression for eta(s) exhibits the correct limiting behaviors. That is, as phi --> 0, the obtained expression for eta(s) becomes that for a dilute suspension of uncharged soft spheres (Ohshima, Langmuir 2008, 24, 6453). As a --> 0, the obtained viscosity expression becomes that for the case of a concentrated suspension of uncharged porous spheres (Ohshima, Colloids Surf. A: Physicochem. Eng. Aspects 2009, 347, 33). In the further limit of a --> 0 and phi --> 0, the obtained viscosity expression becomes the viscosity expression derived for the case of a dilute suspension of uncharged porous spheres by Natraj and Chen (J. Colloid Interface Sci. 2002, 251, 200). As lambda --> infinity or a --> b, the obtained viscosity expression tends to Simha's result (J. Appl. Phys. 1952, 23, 1020) for a concentrated suspension of uncharged rigid spheres of radius b, while as lambda --> 0, it becomes that for uncharged rigid spheres of radius a.

Primary electroviscous effect in a dilute suspension of soft particles

A theory for the primary electroviscous effect in a dilute suspension of soft particles (i.e., particles coated with an ion-penetrable surface layer of polyelectrolytes) in an electrolyte solution is presented. The general expression for the effective viscosity eta s of the suspension and the primary electroviscous coefficient p, which is further expressed in terms of a function L, is given. On the basis of the general expressions, we derive approximate analytic expressions for eta s and p, which are applicable when the density of the fixed charges distributed within the surface layer is low. Further we obtain a simple approximate analytic expression (without involving numerical integrations) for p applicable for most practical cases. It is found that the function L exhibits a minimum when plotted as a function of kappa a (kappa is the Debye-Hückel parameter and a is the particle core radius), unlike the case of a suspension of hard particles, in which case L decreases as kappa a increases, exhibiting no minimum. The presence of a minimum for the case of a suspension of soft particles is due to the fact that L is proportional to 1/kappa 2 at small kappa a and to kappa 2 at large kappa a. Because of the presence of this minimum, the difference in L between soft and hard particles becomes very large for large kappa a.

Primary electroviscous effect in a moderately concentrated suspension of charged spherical colloidal particles

On the basis of the standard theory of the primary electroviscous effect in a moderately concentrated suspension of charged spherical particles in an electrolyte solution presented by Ruiz-Reina et al. (Ruiz-Reina, E.; Carrique, F.; Rubio-Hernández, F. J.; Gómez-Merino, A. I.; García-Sánchez, P. J. Phys. Chem. B 2003, 107, 9528), which is applicable for the case where overlapping of the electrical double layers of adjacent particles can be neglected, the general expression for the effective viscosity or the primary electroviscous coefficient p of the suspension is derived. This expression is applicable for a suspension of spherical particles of radius a carrying arbitrary zeta potentials zeta at the particle volume fraction phi < or = 0.3 for the case of nonoverlapping double layers, that is, at kappaalpha > 10 (where kappa is the Debye-Hückel parameter). A simple approximate analytic expression for p applicable for particles with large kappaalpha and arbitrary zeta is presented. The obtained viscosity expression is a good approximation for moderately concentrated suspensions of the particle volume fraction phi < or = 0.3, where the relative error is negligible for kappaalpha > or =100 and even at kappaalpha = 50 the maximum error is approximately 20%. It is shown that a maximum of p, which appears when plotted as a function of the particle zeta potential, is due to the relaxation effect as in the case of the electrophoresis problem.