Coulomb energy of uniformly-charged spheroidal shell systems

TL;DR

This paper derives exact electrostatic energy expressions for uniformly-charged spheroidal shells, analyzes shape-dependent energy variations, and explores counterion condensation effects, revealing conditions favoring spheroidal over spherical structures.

Contribution

It provides the first exact formulas for spheroidal shell energies, investigates shape perturbations, and models counterion effects, advancing understanding of electrostatics in charged shell systems.

Findings

Prolate spheroids with eccentricity > 0.9 have lower energy than spheres of same area.

Spheres have the highest Coulomb energy among spheroidal shells at fixed volume.

Counterion condensation favors spheroidal structures at high volume fractions.

Abstract

We provide exact expressions for the electrostatic energy of uniformly-charged prolate and oblate spheroidal shells. We find that uniformly-charged prolate spheroids of eccentricity greater than 0.9 have lower Coulomb energy than a sphere of the same area. For the volume-constrained case, we find that a sphere has the highest Coulomb energy among all spheroidal shells. Further, we derive the change in the Coulomb energy of a uniformly-charged shell due to small, area-conserving perturbations on the spherical shape. Our perturbation calculations show that buckling-type deformations on a sphere can lower the Coulomb energy. Finally, we consider the possibility of counterion condensation on the spheroidal shell surface. We employ a Manning-Oosawa two-state model approximation to evaluate the renormalized charge and analyze the behavior of the equilibrium free energy as a function of the shell's aspect ratio for both area-constrained and volume-constrained cases. Counterion condensation is seen to favor the formation of spheroidal structures over a sphere of equal area for high values of shell volume fractions.