The Diffuse Layer of Thin Cylinders: Beyond the Counterion Condensation Paradigm

Abstract

For over half a century, the counterion condensation theory has served as the dominant view for highly charged cylinders, prescribing a sharp charge renormalization based on the idealized abstraction of a salt-free line charge. In this work, we resolve this problem for physical open-domain systems across all charge regimes using the method of matched asymptotics applied to the non-linear Poisson--Boltzmann equation with added salt. For subcritical fields, where the dimensionless charge parameter satisfies $\xi <1$, cylindrical curvature suppresses non-linear coupling, leaving the diffuse atmosphere unmodified. Beyond the critical threshold ($\xi =1$), where the linear theory breaks down due to a localized Boltzmann explosion, the counterion accumulation smoothly transitions into a local flat-wall near-field geometry. By matching these zones, we construct a unified, parameter-free composite analytical solution capturing the spatial crossover toward the global cylindrical Debye screening tail. In the supercritical regime ($\xi >1$), this near-field regularization locks the far-field effective charge into distinct, concentration-dependent parallel plateaus. Crucially, within the realistic regime where $\xi \le 5$, these idealized plateaus are not always reached, and the effective charge operates within a smooth, variable transition zone. Throughout all regimes, the ionic atmosphere remains diffuse, proving that the limiting laws of condensation emerge as artifacts of omitting the screening electrolyte.