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.