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,Rm, with respect to the fractal dimension,df, and the fluid penetration depth,delta, have largely remainedunexplored. Herein, we examine these aspects across a wide range ofdf'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 tobe 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, withdfvarying from 1.76 to 3, and the number of monomersranging from 20 to 10 240 are considered. While confirming the expected asymptotic cluster-size independence of the hydrodynamicratio,beta=Rm/Rg(whereRgis the radius of gyration of the cluster), this study reveals a monotonically increasing trend for beta withincreasingdf. The decay of thefluid velocity within the aggregate is quantified via the concept of penetration depth (delta). Analysisshows that the dimensionless penetration depth (delta*=delta/Rg) approaches asymptotic constancy with respect to cluster size in contrastto a weak dependency of the form delta*similar to(Rg/a)-(df-1)/2, predicted by the mean-field theory (abeing the monomer radius).Furthermore, the penetration depth is found to decrease rapidly, in an exponential manner, with increasing beta. This establishes aquantitative relationship between the resistance experienced by the cluster and the degree of penetration of fluid into it. Theimplications of these results are further discussed.