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 ξ< 1, cylindrical curvature suppresses non-linear coupling, leaving the diffuse atmosphere unmodified. Beyond the critical threshold (ξ= 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 (ξ> 1), this near-field regularization locks the far-field effective charge into distinct, concentration-dependent parallel plateaus. Crucially, within the realistic regime where ξ≤ 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.
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BibTeX
@misc{vinogradova2026diffuse,
title = {{The Diffuse Layer of Thin Cylinders: Beyond the Counterion Condensation Paradigm}},
author = {Olga I. Vinogradova and Elena F. Silkina},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39661},
eprint = {2610.39661},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39661},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Diffuse Layer of Thin Cylinders: Beyond the Counterion Condensation Paradigm
AU - Olga I. Vinogradova
AU - Elena F. Silkina
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39661
UR - https://arxiv.org/abs/2610.39661
AB - 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 ξ< 1, cylindrical curvature suppresses non-linear coupling, leaving the diffuse atmosphere unmodified. Beyond the critical threshold (ξ= 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 (ξ> 1), this near-field regularization locks the far-field effective charge into distinct, concentration-dependent parallel plateaus. Crucially, within the realistic regime where ξ≤ 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.
ER -