Speciation dynamics of metals in dispersion of nanoparticles with discrete distribution of charged binding sites

Abstract

We report a comprehensive theory to evaluate the kinetics of complex formation between metal ions and charged spherical nanoparticles. The latter consist of an ion-impermeable core surrounded by a soft shell layer characterized by a discrete axisymmetric 2D distribution of charged sites that bind metal ions. The theory explicitly integrates the conductive diffusion of metal ions from bulk solution toward the respective locations of the reactive sites within the particle shell volume. The kinetic constant k^os_a for outer-sphere nanoparticle–metal association is obtained from the sum of the contributions stemming from all reactive sites, each evaluated from the corresponding incoming flux of metal ions derived from steady-state Poisson–Nernst–Planck equations. Illustrations are provided to capture the basic intertwined impacts of particle size, overall particle charge, spatial heterogeneity in site distribution, type of particle (hard, core–shell or porous) and concentration of the background electrolyte on k^os_a. As a limit, k^os_a converges with predictions from previously reported analytical expressions derived for porous particles with low and high charge density, cases that correspond to coulombic and mean-field (smeared-out) electrostatic treatments, respectively. The conditions underlying the applicability of these latter approaches are rigorously identified in terms of (i) the extent of overlap between electric double layers around charged neighbouring sites, and (ii) the magnitude of the intraparticulate metal concentration gradient. For the first time, the proposed theory integrates the differentiated impact of the local potential around the charged binding sites amidst the overall particle field, together with that of the so-far discarded intraparticulate flux of metal ions.