Kinetics of two-step nucleation of crystals

When the nucleation of a stable crystalline phase directly in a supersaturated old phase is greatly retarded, the crystal nuclei might nucleate within faster-forming particles of an intermediate phase. Here we present a theoretical investigation of the kinetics of this two-step nucleation of crystals and derive general expressions for the time dependence of the number of crystals nucleated within the particles of the intermediate phase. The results reveal that crystal nucleation can be strongly delayed by the slow growth of the particles and/or by the slow nucleation of the crystals in them. Furthermore, the linear part of the time dependence of the number of nucleated crystals is determined by the formation rate of the intermediate particles. This is in contrast with the one-step nucleation of crystals when this linear part is determined by the rate of crystal nucleation directly in the old phase. Criteria are proposed for distinction between the one- and two-step nucleation mechanisms, based on the supersaturation dependence of the delay time for nucleation. The application of the theoretical approach to the analysis of experimental data on the nucleation of crystals and other ordered aggregates of protein and other soluble materials is discussed.