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） =sin...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）=1/2ey-2/（ka）ey-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 =ey0/2 +{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.展开更多
文摘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）=1/2ey-2/（ka）ey-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 =ey0/2 +{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.
