Kinetics of fragmentation

Abstract

Abstract. A general discussion of the kinetics of continuous, irreversible fragmentation processes is presented. For a linear process, where particle breakup is driven by an external force, we develop a scaling theory to describe the evolution of the cluster size distribution. We treat the general case where the breakup rate of a cluster of mass x varies as xi. When I > 0, corresponding to larger clusters more likely to break up, the scaled cluster size distribution, $(x), decays with the scaled mass, x, as x-* exp(-ax"), as x -+ 00. For small mass, $(x) has the log-normal form, exp(-a In2 x), if the breakup kernel has a small-size cutoff, while $(x) has a power-law tail in the absence of a cutoff. We also show that a conventional scaling picture applies only for the case I > 0. For I < 0, we develop an alternative formulation for the cluster size distribution, in which the typical mass scale is determined by the initial condition. In this regime, we also investigate the nature of a 'shattering' transition, where mass is lost to a 'dust' phase of zero-mass particles. We also study the kinetics of a nonlinear, collision-induced fragmentation process. We analyse the asymptotic behaviours of a simple-minded class of models in which a two-particle collision results in either: (1) both particles splitting into two equal pieces, (2) only the larger particle splitting in two, or (3) only the smaller particle splitting. We map out the kinetics of these models by scaling arguments and by analytic and numerical solutions of the rate equations. Scaling is found to hold for different ranges of homogeneity index for the three models.

Shattering transitions in collision-induced fragmentation

“…In other cases including grinding processes, explosions in an enclosed volume, and breakup of eddies in a turbulent flow [16], interactions between fragments are essential. Such fragmentation processes are intrinsically nonlinear [17,18,19,20]. In this study, we show that the nature of the mass distribution changes qualitatively due to nonlinearities.…”

“…In terms of the collision counter, the process is linear, d dτ c n = 2c n−1 − c n , and subject to the monodisperse initial conditions c n (0) = δ n,0 , the exact solution is the Poissonian density [18] …”

Microplastics Are Contaminants of Emerging Concern in Freshwater Environments: An Overview

“…These models describe the distribution of fragment sizes that result from breakup events. These processes can be expressed by rate equations that assume each particle is exposed to an average environment, mass is the unit used to characterise a particle, and the size distribution is taken to be spatially uniform [69,70]. These processes can be described linearly (i.e.…”

“…particle breakup is driven only by a homogeneous external agent) or nonlinearly (i.e. additional influences also play a role), and particle shape can be accounted for by averaging overall possible particle shape [69]. The models used to describe these degradation process are often frequently complicated, but as a general rule focus on chain scission in the polymer backbone through (a) random chain scission (all bonds break with equal probability) characterised by oxidative reactions; (b) scission at the chain midpoint dominated by mechanical degradation; (c) chain-end scission, a monomer-yielding depolymerisation reaction found in thermal and photodecomposition processes; and (d) in terms of inhomogeneity (different bonds have different breaking probability and dispersed throughout the system) [71][72][73].…”

Asymptotic behaviour of liquid–liquid dispersions

“…Such a breakage mode has been contemplated in physical literature (cf. [7,8,29]), butat least to the author's knowledge-only its discrete version has been investigated mathematically so far (see [20]). …”