Electrostatic potential between charged particles at an oil-water interface

Electrostatic interactions between point charges embedded into interfaces separating dielectric media are omnipresent in soft matter systems and often control their stability. Such interactions are typically complicated and do not resemble their bulk counterparts. For instance, the electrostatic potential of a point charge at an air-water interface falls off as r^{-3}, where r is the distance from the charge, exhibiting a dipolar behavior. This behavior is often assumed to be generic, and is widely referred to when interpreting experimental results. Here we explicitly calculate the in-plane potential of a point charge at an interface between two electrolyte solutions with different, finite dielectric permittivities and Debye screening lengths, such as oil and water. We show that the asymptotic behavior of this potential is neither a dipole, which characterizes the potential at air-water interfaces, nor a screened monopole, which describes the bulk behavior in a single electrolyte solution. By considering the same problem in arbitrary dimensions, we find that the physics behind this difference can be traced to the asymmetric propagation of the interaction in the two media. Our results should be relevant to understand the effective potential acting between interfacial proteins in biofilms, and the self-assembly of charged colloids at droplet surfaces in oil-water emulsions.