A population balance model for the kinetics of covalent organic framework synthesis

This study presents a population balance model for the kinetics of nucleation and growth in covalent organic framework (COF) synthesis. The model incorporates second-order nucleation and first-order growth rates, consistent with proposals in the literature. Despite having non-linear terms, an implicit analytic solution is derived and then converted to explicit solutions for the monomer concentration and size distribution of COF flakes as a function of time. For experimental definitions of the induction time and the initial growth rate based on yield (y) vs time (t) curves, the model predicts power-law relationships: $tind=0.409kN−1/3kG−2/3cA0−1$ and $dy/dtmax=0.965kN1/3kG2/3cA0$⁠, respectively. We discuss the implications for the interpretation of Arrhenius plots. We also discuss key discrepancies with experiments, including the predicted attainment of 100% yield instead of 30%–40% as observed and the value of the yield at the inflection point in the yield vs time curve. We suggest extensions to the model, including nucleation and growth kinetics with equilibrium solubility limitations and two-dimensional nucleation for the formation of multilayer COF particles.