Abstract: The electrical potential distribution for a charged surface in an electrolyte solution at equilibrium is described by the Poisson-Boltzmann equation. For spherical particle, it is (d²y)/(dX²)+²/X(dy)/(dX) =sinhy, where y is a normalized electrostatic potential, defined as y=eψ/(kT), and ψ is the electrostatic potential. X is a normalized distance from the sphere center with radius a. X=ka+kx=ka+ξ. In this paper a flat-plate approximation method is proposed for the resolution of the PB equation. By using the extended Langmuir's method, PB equation is changed to (d²y)/(dζ²)=1/2e^y-2/(ka)√e^y-1. Performing the integration we obtain the relationship between the surface charge density and surface potential for a spherical colloidal particle with a high surface potential. I=-(dy/dζ)_<sup>ζ=0 =e^y_0/2 +{4/(ka)}. Thus the surface excess of co-ions and the double-layer free energy are easily derived. The success of the flat-plate approximation depends so strongly on the value of surface potential y₀ and the radius of curvature of the spherical particle. When the surface potential increases even if the radius of curvature is relatively small, the flat-plate approximation is also satisfactory approximations for the sphere. It explains why the present expressions are applicable to spherical particles with a high surface potential. These expressions are shown to be satisfactory approximations to exact numerical values.

Key words: Spherical colloidal particle, Double electrical layer, Relationship between surface charge density and surface potential