Exploring the electrical potential inside cylinders beyond the Debye-Hückel approximation: a computer code to solve the Poisson-Boltzmann equation for multivalent electrolytes

International audienceThe electrical potential at the interface between mineral and water is traditionally computed from the Poisson–Boltzmann (P-B) equation. Nevertheless, this partial differential equation is nonlinear and has no analytical solution for cylindrical geometries for instance. For that reason, the linearized P-B equation is mostly used in the literature. In our study, we present a short and easy-to-handle Matlab® code to solve the full (i.e. non-linearized) P-B equation inside a cylinder. Electrical potentials inside silica and montmorillonite nanotubes, containing NaCl or CaCl2 electrolytes, are computed. The zeta potential, which is an input parameter of our code, is first predicted from basic Stern and extended Stern models. We show that the linearized P-B equation overestimates the electrical potential from the full P-B equation when the zeta potential magnitude is above ∼ 25.7 mV at 25° C, especially for Ca2+ ion in solution. This effect increases when salinity decreases from 0.1 to 0.001 mol L−1 because of increasing zeta potentials, and when the cylinder radius decreases to the nanometric range because of overlapping diffuse layers. Our results may have strong implications for simulating the electrical properties of sandstones and clays to interpret their self-potential and complex resistivity response