Metal Speciation Dynamics in Dispersions of Soft Colloidal Ligand Particles under Steady-State Laminar Flow Condition

Abstract

A theory is presented for metal speciation dynamics in a swarm of soft, spherical core-hell colloidal ligand particles under steady-state laminar flow condition. Mass transfer and subsequent complexation of metal species within the reactive, permeable particle shell are governed by the interplay between (i) convective-diffusion of free metal ions M within and around the shell where ligands L are distributed, and (ii) kinetics of ML complex formation/dissociation in the shell. The local concentrations of metal M and complex ML are determined by the convective-diffusion equations with appropriate chemical source term and full account of radial and angular concentration polarization contributions. The steady-state flow field is determined from the solution of Navier-tokes equation including convective acceleration term for the fluid external to the particle, and from Brinkman equation for the internal fluid flow. The confined location of ligands within the particle shell leads to ML formation/dissociation rate constants (denoted as ka* and kd*, respectively) that differ significantly from their counterparts (ka and kd) defined for homogeneous ligand distribution throughout the solution. The relationship between ka,d* and ka,d is derived from the numerical evaluation of the spatial, time-dependent distributions of free and bound metal within and/or outside the particle. The dramatic dependence of ka,d* on hydrodynamic particle softness, Péclet number, soft surface layer thickness, and particle radius are analyzed in the steady-state nonequilibrium chemical regime within the context of dynamic features for colloidal complexes. The analysis covers the limiting cases of hydrodynamically impermeable, hard particles where binding sites are located at the very surface of the particle core (e.g., functionalized latex colloids) and free draining, polymeric ligand particles devoid of a hard core (e.g., porous gel particles). The formalism further applies to any values of the Péclet number, that is, for speciation dynamics determined by kinetic processes coupled to diffusion and/or convection metal mass transfer(s). A discussion is provided for the comparison between the exact numerical results and the analytical formulation based on the approximate Levich expression for convective-diffusion metal flux at the surface of hard ligand particles.