Rates of ionic reactions with charged nanoparticles in aqueous media

A theory is developed to evaluate the electrostatic correction for the rate of reaction between a small ion and a charged ligand nanoparticle. The particle is assumed to generally consist of an impermeable core and a shell permeable to water and ions. A derivation is proposed for the ion diffusion flux that includes the impact of the equilibrium electrostatic field distribution within and around the shell of the particle. The contribution of the extra- and intraparticulate field is rationalized in terms of a conductive diffusion factor, fel, that includes the details of the particle geometry (core size and shell thickness), the volume charge density in the shell, and the parameters defining the electrostatic state of the particle core surface. The numerical evaluation of fel, based on the nonlinear Poisson–Boltzmann equation, is successfully complemented with semianalytical expressions valid under the Debye–Hückel condition in the limits of strong and weak electrostatic screening. The latter limit correctly includes the original result obtained by Debye in his 1942 seminal paper about the effect of electrostatics on the rate of collision between two ions. The significant acceleration and/or retardation possibly experienced by a metal ion diffusing across a soft reactive particle/solution interphase is highlighted by exploring the dependence of fel on electrolyte concentration, particle size, particle charge, and particle type (i.e., hard, core/shell, and entirely porous particles).