H.P. van Leeuwen

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, f(el), 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 f(el), 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 f(el) on electrolyte concentration, particle size, particle charge, and particle type (i.e., hard, core/shell, and entirely porous particles).

Chemodynamics of soft nanoparticulate metal complexes in aqueous media: basic theory for spherical particles with homogeneous spatial distributions of sites and charges.

A theoretical discussion is presented to describe the formation and dissociation rate constants for metal ion binding by soft nanoparticulate complexants. The well-known framework of the Eigen mechanism for metal ion complexation by simple ligands in aqueous systems is the starting point. Expressions are derived for the rate constants for the intraparticulate individual outer-sphere and inner-sphere association and dissociation steps for the limiting cases of low and high charge densities. The charge density, binding site density, and size of the nanoparticle play crucial roles. The effects of the electrostatic potential and particle radius on the overall complexation reaction are compared with those for simple ligands. The limitations of the proposed approach for nanoparticulate ligands are discussed, and key issues for future developments are identified.

Metal Speciation Dynamics in Monodisperse Soft Colloidal Ligand Suspensions

A comprehensive theory is presented for the dynamics of metal speciation in monodisperse suspensions of soft spherical particles characterized by a hard core and an ion-permeable shell layer where ligands L are localized. The heterogeneity in the binding site distribution leads to complex formation/dissociation rate constants (denoted as k a (*) and k d (*), respectively) that may substantially differ from their homogeneous solution counterparts (k a and k d). The peculiarities of metal speciation dynamics in soft colloidal ligand dispersions result from the coupling between diffusive transport of free-metal ions M within and around the soft surface layer and the kinetics of ML complex formation/dissociation within the shell component of the particle. The relationship between k a,d (*) and k a,d is derived from the numerical evaluation of the spatial, time-dependent distributions of free and bound metal. For that purpose, the corresponding diffusion equations corrected by the appropriate chemical source term are solved in spherical geometry using a Kuwabara-cell-type representation where the intercellular distance is determined by the volume fraction of soft particles. The numerical study is supported by analytical approaches valid in the short time domain. For dilute dispersions of soft ligand particles, it is shown that the balance between free-metal diffusion within and outside of the shell and the kinetic conversion of M into ML within the particular soft surface layer rapidly establishes a quasi-steady-state regime. For sufficiently long time, chemical equilibrium between the free and bound metal is reached within the reactive particle layer, which corresponds to the true steady-state regime for the system investigated. The analysis reported covers the limiting cases of rigid particles where binding sites are located at the very surface of the particle core (e.g., functionalized latex colloids) and polymeric particles that are devoid of a hard core (e.g., polysaccharide macromolecules, gel particles). For both the transient and quasi-steady-state regimes, the dependence of k a,d (*) on the thickness of the soft surface layer, the radius of the hard core of the particle, and the kinetic rate constants k a,d for homogeneous ligand solutions is thoroughly discussed within the context of dynamic features for colloidal complex systems.