The Casimir Interaction between Spheres Immersed in Electrolytes

Abstract

We investigate the Casimir interaction between two dielectric spheres immersed in an electrolyte solution. Since ionized solutions typically correspond to a plasma frequency much smaller than kBT/ħ at room temperature, only the contribution of the zeroth Matsubara frequency is affected by ionic screening. We follow the electrostatic fluctuational approach and derive the zero-frequency contribution from the linear Poisson-Boltzmann (Debye-Hückel) equation for the geometry of two spherical surfaces of arbitrary radii. We show that a contribution from monopole fluctuations, which is reminiscent of the Kirkwood-Shumaker interaction, arises from the exclusion of ionic charge in the volume occupied by the spheres. Alongside the contribution from dipole fluctuations, such monopolar term provides the leading-order Casimir energy for very small spheres. Finally, we also investigate the large sphere limit and the conditions for validity of the proximity force (Derjaguin) approximation. Altogether, our results represent the first step towards a full scattering approach to the screening of the Casimir interaction between spheres that takes into account the nonlocal response of the electrolyte solution.