A mesoscopic model of nucleation and Ostwald ripening/stepping: Application to the silica polymorph system

Abstract

Precipitation is modeled using a particle size distribution (PSD) approach for the single or multiple polymorph system. A chemical kinetic-type model for the construction of the molecular clusters of each polymorph is formulated that accounts for adsorption at a heterogeneous site, nucleation, growth, and Ostwald ripening. When multiple polymorphs are accounted for, Ostwald stepping is also predicted. The challenge of simulating the 23 order of magnitude in cluster size (monomer, dimer, …, 1023-mer) is met by a new formalism that accounts for the macroscopic behavior of large clusters as well as the structure of small ones. The theory is set forth for the surface kinetic controlled growth systems and it involves corrections to the Lifshitz–Slyozov, Wagner (LSW) equation and preserves the monomer addition kinetics for small clusters. A time independent, scaled PSD behavior is achieved both analytically and numerically, and the average radius grows with Rave∝t1/2 law for smooth particles. Applications are presented for the silica system that involves five polymorphs. Effects of the adsorption energetics and the smooth or fractal nature of clusters on the nucleation, ripening, and stepping behavior are analyzed. The Ostwald stepping scenario is found to be highly sensitive to adsorption energetics. Long time scaling behavior of the PSD reveals time exponents greater than those for the classical theory when particles are fractal. Exact scaling solutions for the PSD are compared with numerical results to assess the accuracy and convergence of our numerical technique.