Influence of cell-model boundary conditions on the conductivity and electrophoretic mobility of concentrated suspensions

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 Kuwabaraʹs hydrodynamic boundary conditions, but there are different possible approaches to the electrostatic boundary conditions. We will call them Levine–Nealeʹs (LN, they are Neumann type, that is they specify the gradient of the electrical potential at the boundary), and Shilov–Zharkikhʹs (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), δni (r = b). It is demonstrated that the conditions δni(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.