Accurate and fast numerical algorithms for tracking particle size distributions during nanoparticle aggregation and dissolution†
Particle size affects the toxicity and environmental fate of engineered nanoparticles (NPs). Although size effects have been widely studied in the experimental NP fate and toxicity literature, no numerical models published to date have attempted to identify and compare state-of-the-art particle modeling methods commonly applied in closely related fields. We compare four numerical frameworks for modeling changes in the size distribution of a NP suspension undergoing dissolution and aggregation: the Sectional Method (SM), Direct Simulation Monte Carlo (DSMC), the Direct Quadrature Method of Moments (DQMOM), and the Extended Quadrature Method of Moments (EQMOM). The SM and the DQMOM were faster or more accurate than the EQMOM and DSMC in nearly every trial. For cases simulating aggregation, the DQMOM took seconds to achieve solutions with ≤2% error, while the SM (a rigorous implementation of the most popular population balance method to date for NPs) took up to 1.5 hours. The SM was far more accurate than the DQMOM for dissolution test cases; however, up to 50 size bins were required to achieve ≤10% error. This raises questions about the validity of the current practice of using five or fewer bins in models of NP fate in rivers. Because runtimes become prohibitive as environmental complexity and particle properties are added, the DQMOM is promising for field-scale models and models that describe NPs with complex morphologies or compositions, such as non-spherical NPs, coated NPs, and nanohybrids. MATLAB code for all models is provided in the ESI.