The method of moments used in polymerization reaction engineering for 70 years: An overview, tutorial, and minilibrary

In the last seven decades, the method of moments (MoM) has become an invaluable tool in the field of polymerization reaction engineering, due to the simplicity of translating a complex set of population balance equations (PBE) into a system with a limited number of equations. In this work, we offer an overview of the MoM, describing the derivation of the moment equations in a basic kinetic mechanism. Some tools and strategies for the derivation of the moment equations are reviewed and explained in detail, such as the binomial theorem, the method of series expansion and pattern identification (SEPI), extensively used by the community, and a graphical approach of summation inversion. The treatment for multivariate distributions is also exposed, taking advantage of the complete and partial moment techniques. The derivation of the MoM contribution by kinetic mechanisms beyond the basic ones or involving special difficulties, such as depropagation, long chain branching (LCB), random chain scission, LCB and ss-scission, short chain branching (SCB) and scission, internal double bond (IDB) (polymerization), termination by combination, reversible deactivation radical polymerization (RDRP), and intermolecular transesterification reactions (ITRs), are explained in a tutorial way. Additionally, the fundamentals of the MoM in copolymerization and the application of the pseudo-homopolymerization approach are briefly described. An introduction of the MoM to emulsion polymerization is also presented. Finally, some advanced applications of MoM in recent works are exposed: MoM models with chain-length dependent or diffusion-controlled termination, and the extension of the MoM for the prediction of the molecular weight distribution (MWD).

Automated summary

In the field of polymerization reaction engineering, the method of moments (MoM) has been widely used due to its simplicity in translating complex population balance equations (PBE) into a limited number of equations. This work provides an overview of the MoM, covering the derivation of moment equations in a basic kinetic mechanism and discussing various tools and strategies such as the binomial theorem, series expansion, pattern identification, and graphical summation inversion. The article also explains the application of MoM in different types of reactions and presents recent advancements including models with chain-length dependent termination and prediction of molecular weight distribution (MWD) using MoM.