Anomalous multipole expansion: Charge regulation of patchy inhomogeneously charged spherical particles

Charge regulation is an important aspect of electrostatics in biological and colloidal systems, where the charges are generally not fixed but depend on the environmental variables. Here, we analyze the charge regulation mechanism in patchy inhomogeneously charged spherical particles, such as globular proteins, colloids, or viruses. Together with the multipole expansion of inhomogeneously charged spherical surfaces, the charge regulation mechanism on the level of linear approximation is shown to lead to a mixing between different multipole moments depending on their capacitance-the response function of the charge distribution with respect to the electrostatic potential. This presents an additional anomalous feature of molecular electrostatics in the presence of ionic screening. We demonstrate the influence of charge regulation on several examples of inhomogeneously charged spherical particles, showing that it leads to significant changes in their multipole moments.