Ostwald ripening with size-dependent rates: Similarity and power-law solutions

The dynamics of Ostwald ripening is treated by cluster distribution kinetics represented by a population balance equation that also describes growth or dissolution. Unlike simple crystal growth driven by supersaturation, the smaller, more soluble clusters in the distribution dissolve during ripening near equilibrium and vanish when they reach the critical nucleus size. Larger clusters accordingly grow as the supersaturation decreases. The long-time asymptotic result of the numerical solution of the scaled population balance equation is power-law decrease of cluster number and growth of average cluster mass, C(u). The cluster distribution approximates an exponential self-similar solution, and eventually narrows until but one large cluster remains, satisfying the mass balance. A previous theory is here extended to include mass-dependent rate coefficients for growth and dissolution that satisfy microscopic reversibility. The asymptotic power-law growth, C ;u, is determined by the power l on the mass for rate coefficients. The power is l51/3 for diffusion-controlled and l52/3 for surface-controlled processes. Experimentally observed ripening behavior is realized by an apt choice of l for a given time range.