Potential of a sphere in an ionic solution in thin double layer approximations

Abstract

The three-dimensional problem of a charged spherical particle immersed in an ionic solution is solved asymptotically. The small parameter used is δ/a, where δ is the Debye screening distance and a the radius of the particle. The solutions to the Poisson–Boltzmann equation are obtained inside the double layer and outside the double layer. They are matched together asymptotically to obtain a solution whose error is of order δ2/a2 smaller than the expression kept. A uniform solution valid in all the regions is also obtained. Comparison with numerical solution to the problem, which is usually tedious, shows excellent agreement.