Asphericity Lifts the Degeneracy Among Ordering Mechanisms on Curved Surfaces

Abstract

Spherical surfaces conceal the one-body signature of particle organization: area-uniform, conductor, and curvature-weighted measures coincide there, so the ordering mechanism leaves no trace in the density. Aspherical deformation lifts this degeneracy. On an ellipsoid, these ordering mechanisms separate into distinct powers of the distance from the shell center to its tangent plane. To leading order in deformation, the characteristic exponent can be read from both the structure factor quadrupole and the radius of gyration. Two interaction families realize the exponent in complementary ways: the Riesz $s$-energy, whose limiting measure switches sharply at the marginal power, and screened Coulomb potential, where the exponent is tuned continuously by the salt concentration. Ionic strength thereby becomes an experimental dial that transforms the particle ordering across the conductor-to-packing crossover on a fixed aspherical surface, turning a macroscopic shape parameter into a probe of the microscopic interaction law.