A rapidly convergent series for calculation of the interaction between two similar plane double-layers for z+/z−=−2z+/z−=−2 asymmetric electrolytes at positive surface potential

A rapidly convergent series for calculation of the interaction energies between two similar plane double-layers for z+/z−=−2z+/z−=−2 asymmetric electrolytes at positive surface potential are obtained by introducing a parameter λ   in elliptical integral. When dimensionless surface potential is less or equal to 20, the number of the series terms required to obtain the interaction energies with six significant digits are not more than 4. The accurate numeral results are given and they can be used to check up the validity of approximate expressions people obtain. The present results are also fit for z+/z−=−1/2z+/z−=−1/2 asymmetric electrolytes at negative surface potential.

Graphical abstractVz−z′ and V′V′ in the figure are the interaction energies of symmetric and z+/z−=−2z+/z−=−2 asymmetric electrolytes at y>0y>0, respectively. Vz−z′ must intersect V′V′ at a certain ξdξd. When ξdξd is smaller V′>Vz−z′; when ξdξd is larger, V′