Multipole expansion of the electrostatic interaction between charged colloids at interfaces

The general form of the electrostatic potential around an arbitrarily charged colloid at a flat interface between a dielectric and a screening phase (such as air and water, respectively) is analyzed in terms of a multipole expansion. The leading term is isotropic in the interfacial plane and varies with d(-3) where d is the in-plane distance from the colloid. The effective interaction potential between two arbitrarily charged colloids is likewise isotropic and proportional to d(-3), thus generalizing the dipole-dipole repulsion first found for point charges at water interfaces. Anisotropic attractive interaction terms can arise only for higher powers d(-n) with n >= 4. The relevance of these findings for recent experiments is discussed.