Comparison of a new mass-concentration, chain-reaction model with the population-balance model for early- and late-stage aggregation of shattered graphene oxide nanoparticles

Aggregation as an essential mechanism impacting nanoparticle (NP) functionality, fate, and transport in the environment is currently modelled using population-balance equation (PBE) models which are computationally expensive when combined with other continuum-scale reactive transport models. We propose a new simple mass-concentration-based, chain-reaction modelling (CRM) framework to alleviate computational expenses of PBE and potentially to facilitate combination with other fate, transport, and reaction models. Model performance is compared with analytical PBE solution and a standard numerical PBE technique (fixed pivot, FP) by fitting against experimental data (i.e., hydrodynamic diameter and derived count rate of dynamic light scattering used as a representative of mass concentration) for early- and late-stage, aggregation of shattered graphene oxide (SGO) NP across a broad range of solution chemistries. In general, the CRM approach demonstrates a better match with the experimental data with a mean Nash-Sutcliffe model efficiency (NSE) coefficient of 0.345 than the FP model with a mean NSE of 0.29. Comparing model parameters (aggregation rate constant and fractal dimension) obtained from fitting CRM and FP to the experimental data, similar trends or ranges are obtained between the two approaches. Computationally, the modified CRM is an order-of-magnitude faster than the FP technique, suggesting that it can be a promising modelling framework for efficient and accurate modelling of NP aggregation. However, in the scope of this study, reaction rate coefficients of the CRM have been linked to collision frequencies based on simplified and empirical relationships which need improvement in future studies.