New solution of the non-linear Poisson-Boltzmann differential equation for solid particle dispersions in dissymmetrical electrolytes

A new mathematical solution to the non-linear Poisson-Boltzmann differential equation for solid-liquid dispersions in presence of different dissymmetrical electrolytes was given. The analytical expressions of the surface and charge density of solid particles were given. The variations of electrostatic potential ψ (x) and charge density σ (x) of dispersed particles against the distance x were obtained. For colloidal particles in presence of E(m-n) electrolytes with $m\ne n$ with $m\ge 3,n\ge 3$ and for E(2–3) and E(3-2) electrolytes, the mean electrostatic potential as a function of the distance was numerically integrated by Mathematica program version 13.

The experimental study of silica suspensions in presence with the following electrolytes $NaCl$, ${Na}_2{SO}_4$, $Ca{Cl}_2$, ${Na}_3{PO}_4$, $Al{Cl}_3$, ${Al}_2{\left({SO}_4\right)}_3$, ${Ca}_3{\left({PO}_4\right)}_2$, ${Na}_4{P_{2}O}_7$ and ${Na}_5{P_{3}O}_{10}$ led to confirm the theoretical predictions obtained from the analytical solution of Poisson-Boltzmann equation. The results obtained allowed to determine the surface potential as a function of pH of the suspension and the electrostatic potential versus the distance x. The variations of the dissociation coefficient of silica surfaces were determined. An important effect of the anion and cation valences of the dissymmetrical electrolytes on the surface charge density and potential was highlighted.