A model of ballistic aggregation and fragmentation

TL;DR

This paper introduces a simple kinetic model for ballistic aggregation and fragmentation based on energy thresholds, analyzing how particles grow or break apart, with solutions derived analytically, numerically, and via simulations.

Contribution

It presents a novel, analytically tractable model incorporating energy thresholds for aggregation and fragmentation, with analysis of different dynamic regimes and mass-dependent effects.

Findings

Unlimited cluster growth occurs with weak fragmentation.

Steady states emerge when fragmentation dominates.

Mass-dependent thresholds cause regime cross-over.

Abstract

A simple model of ballistic aggregation and fragmentation is proposed. The model is characterized by two energy thresholds, Eagg and Efrag, which demarcate different types of impacts: If the kinetic energy of the relative motion of a colliding pair is smaller than Eagg or larger than Efrag, particles respectively merge or break; otherwise they rebound. We assume that particles are formed from monomers which cannot split any further and that in a collision-induced fragmentation the larger particle splits into two fragments. We start from the Boltzmann equation for the mass-velocity distribution function and derive Smoluchowski-like equations for concentrations of particles of different mass. We analyze these equations analytically, solve them numerically and perform Monte Carlo simulations. When aggregation and fragmentation energy thresholds do not depend on the masses of the colliding particles, the model becomes analytically tractable. In this case we show the emergence of the two types of behavior: the regime of unlimited cluster growth arises when fragmentation is (relatively) weak and the relaxation towards a steady state occurs when fragmentation prevails. In a model with mass-dependent Eagg and Efrag the evolution with a cross-over from one of the regimes to another has been detected.