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, 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.

Fractal Growth of Quasi Two-Dimensional Copper Dendrites by Template-free Electrodeposition

“…To describe the growth of irregular dendrites, fractal theory was introduced in 1984 . Theoretical models, such as diffusion-limited aggregation (DLA) and kinetic cluster aggregation, were developed primarily to illustrate the internal physical chemistry mechanism of fractal growth. − According to the DLA model, fractal growth often occurs at a state far from equilibrium, and the random walking of ions is the key to the formation of fractal structure deposits. − But, the unidirectional movement of ions under a flow field or electric field in the electrodeposition process is ignored. Thus, researching the ion movement in the electrochemical process will provide a theoretical basis for the electrochemical assembly of micro-and nanoscale novel materials, which can realize the application of many functional devices. − An excellent photothermal device often has the characteristics of micro–nano structures, mainly concentrated between 10 nm and 500 μm. − CuO is one of the promising photothermal materials because of its high light absorption and good solar selectivity.…”