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

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, R m , with respect to the fractal dimension, d f , and the fluid penetration depth, δ, have largely remained unexplored. Herein, we examine these aspects across a wide range of d f '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 d f 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 = R m / R g (where R g is the radius of gyration of the cluster), this study reveals a monotonically increasing trend for β with increasing d f . 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/ R g ) approaches asymptotic constancy with respect to cluster size in contrast to a weak dependency of the form δ* ∼ ( R g / a ) -( d f  - 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.