Characteristic curves and relationships for surface charging parameters of nonelectrolyte-immersed aqueous cavities

Abstract

Most properties of charged particles’ dispersions are closely related to the amount of charge residing on the surface of particles. Very few works have been done so far to investigate curvature-dependence of surface potential and charge density of dispersed charged particles. Although there are some experimental studies in the literature on this subject, only a few of them have examined the mathematical relationship between particle size and surface charging parameters, resulting in contradictory findings. Indeed, a universal behavior is yet to be inferred from analysis of the outcomes of these studies since they reported either a roughly constant or decreasing trend of surface charge density with increasing particle radius. Unfortunately, the majority of the limited research efforts on surface charging have focused on dispersion of charged particles in aqueous solutions, but there are systems in biological and engineering worlds where an electrolyte solution is encapsulated in the core of charged particles and immersed in a background non-electrolyte medium. In an attempt to fill this scientific gap and gain a better understanding of surface charging, analytical models are developed in the present work for prediction of surface potential, surface charge, and surface charge density for cylindrical and spherical particles in such systems.First, the problem under investigation is non-dimensionalized to facilitate the modeling process, consider the simultaneous effects of key variables, and generalize the results and behaviors. Then, the mutual relationship between every pair of variables is extracted by employing Gauss's law and combining it with the electro-neutrality condition in an aqueous cavity. Afterwards, the resulting generalized seemingly simple, but inherently complex, integral equations are solved to build the characteristic curves and regressed to derive explicit characteristic equations. Finally, the derived characteristic equations are transformed into a dimensional form to obtain simple formulas for explicit calculations of charging parameters on the surface of electrolyte-encapsulating charged particles in terms of easily measurable properties of the electrolyte solution, the size of the cavity, and thermodynamic conditions. This predictive capability of the new models will significantly reduce the time, costs, and efforts required to simultaneously characterize the size and charging parameters of particles because it removes the need for such a challenging measurement as charging parameters are explicitly related to particle size through thermodynamic conditions, as well as available or easy-to-measure properties of the solution.