Coulomb energy of uniformly charged spheroidal shell systems

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.

Computational studies of shape control of charged deformable nanocontainers

“…By virtue of the electrostatics-driven deformation, environmental (solutionbased) control strategies (e.g., tuning salt concentration) as well as materials design approaches (e.g., tuning surface charge) were demonstrated as a feasible means of controlling container shape and changing preferences for bowl-like or disc-like morphologies. Effects of ion condensation were analyzed using a two-state Manning mean-field model, which supported the feasibility of the observed shape transitions 39,44 .…”

“…Early work on charged elastic membranes (shells) mainly focused on mapping a charged, deformable membrane to an uncharged, deformable membrane with charge-renormalized elastic parameters [40][41][42] . Recent studies have modeled Coulomb interactions explicitly to accurately describe the nonlinear coupling between elastic and electrostatic forces in charged nanoscale containers where the surface charge density can assume high values ( 0.1e/ nm 2 ) 38,43,44 . Using such models, it has been shown that an ionic shell, where positive and negative charges populate the surface, lowers its energy by taking a faceted (icosahedral) shape with the same surface area 43 .…”

“…Building on our previous studies, here we design a minimal coarse-grained model of a flexible nanomembrane representative of monolayer vesicles and protein nanocages in electrolyte solution to establish the links between surface properties of uniformly-charged, volume-conserving deformable nanocontainers and their equilibrium shapes by using molecular dynamics (MD) simulations 39,44 . We systematically explore the design space generated by the nanocontainer size, surface charge, bending rigidity κ b , stretching constant κ s , and the bulk solution salt concentration and interfacial surface tension to show that flexible nanocontainers of radii ranging from 10 -20 nm exhibit sphere-to-rod-to-disc shape transitions.…”

Electrostatic interactions between spheroidal dielectric particles

“…29 However, there have not been corresponding developments towards a general methodology for treating electrostatic interactions between non-spherical particles. Exact solutions to this problem have only been presented for a single uniformly charged spheroidal shell, 30 and where the image charge method has been used to treat conducting ellipsoidal particles. 31 In this paper, an analytical theory of electrostatic interactions between spheroidal particles has been developed, building on previous work, [32][33][34][35][36] where analytical expressions have been given for the electrostatic force between charged, dielectric sphere -sphere 32 and sphereplanar surface systems.…”

Screened Coulomb interactions of general macroions with nonzero particle volume

“…For nonspherical shapes, the screened-electrostatic pair interaction is only analytically known in a few cases for all particle configurations, even within linear screening theory. However, some studies exist for disks [35][36][37][38], rods [39,40], spheroids [41][42][43][44][45], or helices [46], where the potential is sometimes calculated only for infinitely long, thin, or ion-penetrable particles, restricted particle configurations, or orientation-averaged interactions [47]. The difficulty in finding analytical solutions lies in the finite ion-impenetrable particle volume which complicates matching the series expansion solution (if it is even available for the geometry under consideration) of the unscreened potential inside and the screened potential outside the particle via the boundary conditions.…”