The accuracy of the Derjaguin approximation for the electrostatic double-layer interaction between curved surfaces bearing constant potentials

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Abstract
This study examines the validity of the Derjaguin approximation often used to obtain the electrostatic double-layer (e.s.d.l.) force between curved surfaces from the mathematically simpler e.s.d.l. interaction energy between parallel plates. Here one uses an exact solution for the force between spheres of unequal size bearing constant but unlike surface potentials. One compares the planar interaction energy computed by the approximation with the known exact planar energy. In the limits of large and small separation, the approximation works remarkably well, even at high potentials. However, at intermediate separations, and particularly in regions where force and energy go through extrema, there is a qualitative difference between energies computed exactly and those inferred from the approximation. The reason for this discrepancy appears to be that the force between different parts of the spherical surface changes sign at different separations, while the change in sign of force in the planar case is uniform for all parts of the surface.

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