Mobility and settling rate of agglomerates of polydisperse nanoparticles

Agglomerate settling impacts nanotoxicology and nanomedicine as well as the stability of engineered nanofluids. Here, the mobility of nanostructured fractal-like SiO₂ agglomerates in water is investigated and their settling rate in infinitely dilute suspensions is calculated by a Brownian dynamics algorithm tracking the agglomerate translational and rotational motion. The corresponding friction matrices are obtained using the HYDRO++ algorithm [J. G. de la Torre, G. del Rio Echenique, and A. Ortega, J. Phys. Chem. B 111, 955 (2007)] from the Kirkwood-Riseman theory accounting for hydrodynamic interactions of primary particles (PPs) through the Rotne-Prager-Yamakawa tensor, properly modified for polydisperse PPs. Agglomerates are generated by an event-driven method and have constant mass fractal dimension but varying PP size distribution, mass, and relative shape anisotropy. The calculated diffusion coefficient from HYDRO++ is used to obtain the agglomerate mobility diameter d_m and is compared with that from scaling laws for fractal-like agglomerates. The ratio d_m/d_g of the mobility diameter to the gyration diameter of the agglomerate decreases with increasing relative shape anisotropy. For constant d_m and mean d_p, the agglomerate settling rate, u_s, increases with increasing PP geometric standard deviation σ_p,g (polydispersity). A linear relationship between u_s and agglomerate mass to d_m ratio, m/d_m, is revealed and attributed to the fast Brownian rotation of such small and light nanoparticle agglomerates. An analytical expression for the u_s of agglomerates consisting of polydisperse PPs is then derived, u_s=1-ρ_fρ_pg3πμmd_m (ρ_f is the density of the fluid, ρ_p is the density of PPs, μ is the viscosity of the fluid, and g is the acceleration of gravity), valid for agglomerates for which the characteristic rotational time is considerably shorter than their settling time. Our calculations demonstrate that the commonly made assumption of monodisperse PPs underestimates u_s by a fraction depending on σ_p,g and agglomerate mass mobility exponent. Simulations are in excellent agreement with deposition rate measurements of fumed SiO₂ agglomerates in water.