Charge-Regulated Interactions: The Case of a Nanoparticle and a Sphere of Arbitrary Dielectric Constants

Surfaces of objects immersed in liquid develop an electric charge density that depends on the types and concentrations of dissolved ions. The strength and spatial distribution of this charge density controls a myriad of processes, from biological to industrial processes. In addition, the lack of a full understanding of the charge density precludes a complete foundational interpretation of liquid-mediated many-body interactions. This understanding is especially obscured by charge regulation, whereby the charge on an object hinges, in addition, on the charges and locations of all other charged objects in the liquid. Here, we present a rigorous mathematical approach based on the Poisson-Boltzmann Equation, with field-dependent boundary conditions, and apply it to obtain the liquid-mediated interaction energy between a charged dielectric sphere and a charged particle. The framework that we develop in this article should be of use beyond the limits of the example application considered here: it should be useful as a conceptual and technical starting point to obtain charge-regulated many-body interactions in liquids.