Effective Parameters of Charged Spherical Particles in 1 : 1 Electrolyte Solutions

Abstract

In scope of the Poisson–Boltzmann theory, the electrostatic potential profiles in the vicinity of spherical particles immersed in 1 : 1 electrolyte solutions have been precisely calculated. Using the data on the behavior of the profiles at large distances from the particle surface, effective surface potential $${{\psi }_{{{\text{eff}}}}},$$ and its limiting value $$\psi _{{{\text{eff}}}}^{{{\text{sat}}}},$$ to which it tends upon an infinite growth of the surface charge, have been determined for a wide range of model parameters (surface charge density, particle radius, and electrolyte concentration). A universal curve has been plotted to represent the dependence of $$\psi _{{{\text{eff}}}}^{{{\text{sat}}}}$$ on reduced particle radius κa (a is the radius and κ is the reciprocal Debye screening radius) and to evidently illustrate the existence of two known limiting laws of variations in the effective potential that corresponds to the saturation conditions. The energy criterion and the analysis of its sensitivity to the cutoff threshold have been employed to evaluate the thicknesses of the shells formed by immobilized counterions around the spherical particles. Dependences of the shell thickness on the surface charge density, particle radius, and 1 : 1 electrolyte concentration have been analyzed. It has been revealed that there is limiting thickness $$l_{{{\text{eff}}}}^{{{\text{sat}}}},$$ which is reached upon the infinite growth of the surface charge density. A universal κ $$l_{{{\text{eff}}}}^{{{\text{sat}}}}$$ (κa) curve is presented and compared with the $$\psi _{{{\text{eff}}}}^{{{\text{sat}}}}$$ (κa) curve.