2015/11/04 by Ladislav Šamaj, Ladislav Samaj, Emmanuel Trizac +1 · 8 citations
Chemistry · Engineering · Materials Science · Mathematics · Physics and Astronomy · #Boltzmann equation #Classical mechanics #Cylinder #Electrostatics and Colloid Interactions #Geometry #Ion #Material Dynamics and Properties #Mathematics #Nanopore and Nanochannel Transport Studies #Physics #Poisson–Boltzmann equation #Quantum mechanics #Thermodynamics #cond-mat.soft #cond-mat.stat-mech
paper · pdf · doi:10.1103/physreve.93.012601
published in Physical review. E 93(1), 012601 (American Physical Society)
arxiv created 2015/11/04 · openalex publication_date 2016/01/06 · arxiv updated 2016/03/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We consider a set of identical mobile pointlike charges (counterions) confined to a domain with curved hard walls carrying a uniform fixed surface charge density, the system as a whole being electroneutral. Three domain geometries are considered: a pair of parallel plates, the cylinder, and the sphere. The particle system in thermal equilibrium is assumed to be described by the nonlinear Poisson-Boltzmann theory. While the effectively one-dimensional plates and the two-dimensional cylinder have already been solved, the three-dimensional sphere problem is not integrable. It is shown that the contact density of particles at the charged surface is determined by a first-order Abel differential equation of the second kind which is a counterpart of Enig's equation in the critical theory of gravitation and combustion or explosion. This equation enables us to construct the exact series solutions of the contact density in the regions of small and large surface charge densities. The formalism provides, within the mean-field Poisson-Boltzmann framework, the complete thermodynamics of counterions inside a charged sphere (salt-free system).