Ionization of Semidilute Weak Polyelectrolytes in Equilibrium with a Reservoir

We develop a theoretical framework for describing the ionization of semidilute weak polyelectrolytes in equilibrium with a reservoir that defines the pH and salt concentration. Charged monomers are modeled as planar or cylindrical geometries, and the inter-monomer electrostatic interactions are explicitly addressed based on the nonlinear Poisson–Boltzmann equation and Langmuir adsorption isotherm equation. Asymptotic analysis yields analytical scaling laws for the ionization degree across three distinct regimes, determined by the relative magnitudes of the surface potential and charge fraction. These regimes further correspond to different pH intervals where hydrogen, salt, or hydroxide ions dominate the electrostatic and ionization equilibria. Comparison between planar and cylindrical cell models reveals that curvature amplifies near-surface electric fields and ionic enrichment, resulting in stronger charge screening and smaller densities of ionizable groups. By extending the Henderson–Hasselbalch and Donnan descriptions beyond their applicability limits, this theory provides a generic framework that captures ionization in both open and semi-open systems, bridging the transition from overlapping to non-overlapping electric double layers in semidilute weak polyelectrolytes.