Scaling properties of one-dimensional cluster–cluster aggregation with Lévy diffusion

Abstract

We present a study of the scaling properties of cluster-cluster aggregation with a source of monomers in the stationary state when the spatial transport of particles occurs by Lévy flights. We show that the transition from mean-field statistics to fluctuation-dominated statistics which, for the more commonly considered case of diffusive transport, occurs as the spatial dimension of the system is tuned through two from above, can be mimicked even in one dimension by varying the characteristic exponent, β, of the the Lévy jump length distribution. We also show that the two-point mass correlation function, responsible for the flux of mass in the stationary state, is strongly universal: its scaling exponent is given by the mean field value independent of the spatial dimension and independent of the value of β. Finally we study numerically the two point spatial correlation function which characterises the structure of the depletion zone around heavy particles in the diffusion limited regime. We find that this correlation function vanishes with a non-trivial fractional power of the separation between particles as this separation goes to zero. We provide a scaling argument for the value of this exponent which is in reasonable agreement with the numerical measurements.

Kinetics of aggregation in liquids with dispersed nanoparticles

“…5 Typical SDs in inhomogeneous NP systems have been shown to be well described by single-peaked functions of different forms, ranging from relatively narrow to distinctly broad, usually skewed in the right direction. [31][32][33][34][35][36][37] It has been argued that this equation accurately describes the aggregation of clusters with the fractal structure. Indeed, such asymmetric distributions have experimentally been determined, e.g., for systems organized through the competition between capping and reducing agents, 25 as well as for systems of aggregates that appeared as a result of the incorporation of water, 29 or as a consequence of a two-stage process of the attachment of monomers to NPs and the aggregation of the resulting NPs.…”

“…For instance, the aggregation process of NPs can be investigated using the approach based on the kinetic coagulation Smoluchowski equation, or on its modified or extended versions. [31][32][33][34][35][36][37] It has been argued that this equation accurately describes the aggregation of clusters with the fractal structure. 31 The resulting SD has been shown to exhibit self-similarity for both constant aggregation rates (related to the diffusion limited aggregation) and some homogeneous aggregation rates.…”