Numerical and Analytical Studies of the Electrical Conductivity of a Concentrated Colloidal Suspension

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, φ, zeta potential, , and electrokinetic radius, κa (κ -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 κa f ∞. 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 Dirichlettype boundary conditions.

Properties and electrokinetic behavior of non-dilute colloidal suspensions

“…This condition is also consistent with the Maxwell limit for a suspension of uncharged spheres (Maxwell, 1873), where the conductivity of the solution becomes independent of the zeta potential (Carrique et al, 2001;Cuquejo et al, 2006;Jeffrey, 1973;McKenzie et al, 1978;Ohshima, 1999;Sangani and Acrivos, 1983). A detailed description of this phenomenon is provided in Hunter (2001).…”

“…These models primary use the Kuwabara's cell model but differ by the electrostatic boundary conditions at the outer surface of the cell. Cuquejo et al (2006) recently reported an analytical model that matches well with experimental data for wide range of f and /, but only valid in the limit of thin EDL (large ja). To our knowledge there are still no existing models that accurately predict the conductivity modification of a suspension for arbitrary values of the electrokinetic radius ja.…”

“…To our knowledge there are still no existing models that accurately predict the conductivity modification of a suspension for arbitrary values of the electrokinetic radius ja. In spite of all the theoretical formulations that have been developed so far, we are not aware of any accurate method to predict the variation of electric conductivity of suspensions with volume fraction of particles with thick electric double layers ja < 1 (Cuquejo et al, 2006;Ding and Keh, 2001). Understanding the conductivity variations for small ja regime is especially critical for lab-on-achip and electrophoretic deposition applications.…”

“…Typically, a concentrated suspension in the electrokinetics literature may have volume fractions of greater than 10% (Ahualli et al, 2006;Carrique et al, 2001Carrique et al, , 2002Carrique et al, , 2003aCarrique et al, ,b, 2007Carrique et al, , 2006Cuquejo et al, 2006;Ding and Keh, 2001;Dukhin et al, 1999;Johnson and Davis, 1999;Lee et al, 2001;Midmore and Obrien, 1988;Mizuno et al, 2000;Mooney, 1951;Ohshima, 1999Ohshima, , 2000Perez and Lemaire, 2004;Saville, 1987, 1989). It is common place to present the incremental change in a fluid or particle property (e.g.…”

Can we describe charged nanoparticles with electrolyte theories? Insight from mesoscopic simulation techniques

“…Actually, to describe the case of charged nanoparticles, it is difficult to choose the right conceptual framework: Are suspensions of charged nanoparticles still electrolyte solutions, or should they be viewed through the lens of colloidal theories ? Theories of colloidal conductivity are adapted to the description of suspensions in which the length scales relative to the colloid and the ions of the supporting electrolyte are very different (see for instance [32,33] for static, and [34,35] for frequency dependent measurements). For dispersions containing colloidal particles with a nanometric size -a radius smaller than 10 nm-, such theories do not seem to be relevant.…”