Sigmoidal Nucleation and Growth Curves Across Nature Fit by the Finke–Watzky Model of Slow Continuous Nucleation and Autocatalytic Growth: Explicit Formulas for the Lag and Growth Times Plus Other Key Insights

Abstract

Sigmoidal kinetic curves have been reported for a number of cooperative phenomena in nature. These curves may be fit by purely mathematical functions that, however, do not correspond to any physical model. The 1997 Finke–Watzky (F–W) two-step model of slow, continuous nucleation (A → B, rate constant k1) and fast, autocatalytic growth (A + B → 2B, rate constant k2) provides both a physical model and a mathematical solution. As a result, the F–W two-step kinetic model has been successfully applied to a large number of cooperative phenomena throughout nature that display sigmoidal kinetic curves. Herein, we derive formulas for the first, second, and third derivatives of the concentration of product versus time, [B]t, expressed in terms of the F–W parameters k1, k2, and the initial concentration of monomer, [A]0. Mathematical expressions are then derived for key empirical parameters in sigmoidal curves, including the induction period and (maximum) slope, which are then examined under the limit k1 ≪ k2[A]0 where nucleation and growth are well-separated in time. The resultant seven previously unavailable equations provide a better fundamental and intuitive understanding of sigmoidal curves across nature well-fit by the F–W two-step mechanism.