Double-layer interaction energy for two unequal spheres

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Abstract
The potential distribution surrounding two unequal spheres of any surface potential has been obtained by a finite element solution of the Poisson–Boltzmann equation in bispherical coordinates using constant potential boundary conditions. The electrostatic energy of the two-sphere system was obtained from the potential distribution. This method is completely general. It is not restricted to “plate-like” particles or to small surface potentials. A comparison is made between this method of calculating the double-layer interaction energy and other methods which utilize the parallel-plate solution to the Poisson–Boltzmann equation. Both methods give linear results when log(energy) is plotted against separation. However, this method gives a more negative slope than the one-dimensional methods.