Relationship Between the Mobility of Aggregates and Fluid Penetration Depth Across a Range of Fractal Dimensions Using Stokesian Dynamics.

Abstract

The hydrodynamic behavior of fractal aggregates plays an important role in various applications in industry and the environment, and has been a topic of interest over the past several decades. Despite this, crucial aspects such as the relationship of the mobility radius, Rm, with respect to the fractal dimension, df, and the fluid penetration depth, δ, have largely remained unexplored. Herein, we examine these aspects across a wide range of df's through a Stokesian dynamics approach. It takes into account all orders of monomer-monomer interactions to construct the resistance matrix for the entire cluster, which is assumed to be rigid. Statistical fractals created using algorithms such as diffusion limited aggregation (DLA), cluster-cluster aggregation (CCA), tunable Monte Carlo algorithm, and a deterministic Vicsek fractal, with df varying from 1.76 to 3, and the number of monomers ranging from 20 to 10 240 are considered. While confirming the expected asymptotic cluster-size independence of the hydrodynamic ratio, β = Rm/Rg (where Rg is the radius of gyration of the cluster), this study reveals a monotonically increasing trend for β with increasing df. The decay of the fluid velocity within the aggregate is quantified via the concept of penetration depth (δ). Analysis shows that the dimensionless penetration depth (δ* = δ/Rg) approaches asymptotic constancy with respect to cluster size in contrast to a weak dependency of the form δ* ∼ (Rg/a)-(df-1)/2, predicted by the mean-field theory (a being the monomer radius). Furthermore, the penetration depth is found to decrease rapidly, in an exponential manner, with increasing β. This establishes a quantitative relationship between the resistance experienced by the cluster and the degree of penetration of fluid into it. The implications of these results are further discussed.