Nucleation rate and Gibbs free energy of nucleation of APIs, large molecule, amino acid and inorganic materials in solution at different cooling rates

Abstract

We propose a new mathematical model based on the classical nucleation theory to predict the nucleation rate, kinetic constant, and Gibbs free energy of nucleation using metastable zone width (MSZW) data as a function of solubility temperature. Unlike widely used models by Nývlt, Kubota, and Sangwal, which are limited in capturing the impact of varying cooling rates, the proposed model allows direct estimation of nucleation rates from MSZW data obtained under different cooling conditions. This is particularly advantageous for continuous or semi-batch crystallisation design, where cooling rate is a critical variable. The model has been successfully validated using experimental data from 22 solute–solvent systems, including 10 APIs, one API intermediate, lysozyme, and glycine, as well as 8 inorganic compounds. Predicted nucleation rates span from 10²⁰ to 10²⁴ molecules per m³ s for APIs, and up to 10³⁴ molecules per m³ s for lysozyme, the largest molecule studied. Gibbs free energy of nucleation varies from 4 to 49 kJ mol⁻¹ for most compounds, reaching 87 kJ mol⁻¹ for lysozyme. The model also enables accurate prediction of induction time and key thermodynamic parameters such as surface free energy, critical nucleus size, and number of unit cells—based solely on MSZW data obtained at different cooling rates.