Phase transitions, oscillating and super-cluster states in aggregation and fragmentation kinetics

We consider well defined systems involving various subclasses of aggregation and/or fragmentation processes. These systems distinctly describe 1) the aggregation of ash, soot and raindrops in physical studies 2) oscillatory behaviour in aggregation-fragmentation activities and 3) the coagulation (addition) and disintegration (complete or incomplete) of clusters. Each system involves formulating nonlinear differential equations (ordinary or partial) which describe the physical processes at hand. In addition to that, kinetic rates which convey how clusters merge and/or fragment are applied and conditions present at the start of the event are set. The universal task is to analytically and/or numerically study the behaviour of each system for theoretical and/or physical applications.

We begin by first microscopically deriving the rate coefficients of space - nonhomogeneous Smoluchowski equations by considering rainfall coagulation. We assume raindrops fall at the rainfall speed Uk = ̶ u0kµ and monodisperse initial conditions are administered. Results show that the total size distributions N(z; t) and single droplets c1(z; t) monotonically decline to the equilibrium state at each height z. For other raindrop size distributions ck(z,t) (k > 2), ck(z, t) either monotonically/nonmonotonically reach the equilibrium state.

Next, we study the oscillatory behaviour in aggregation-fragmentation systems with aggregation kernel Kij = i ͮ j ͧ + j ͮ i ͧ and fragmentation kernel Fij = λKij (according to Matveev et al. [1, 2]). We find that oscillations occur for the condition v ̶ µ > 1. A disintegration rate λc(v, µ) exists such that λ < λc(v, µ) yields never ending oscillations while decaying oscillations occur when λ > λc(v, µ). For other conditions v ̶ µ < 1, an analytic approximation involving the generating function (and other techniques) is used to find the steady state solution.

Lastly, we investigate the process of addition with disintegration. We first consider systems involving monomer-cluster interactions with power law dependent addition rate Ak = kª and shattering rate Sk = λ ks. A cluster grows by an addition of a monomer onto the cluster and completely shatters into monomers when an energetic monomer collides into it. The interplay between addition and shattering leads to the existence of phase transitions which separate different evolutionary behaviour. The phase transition can be discontinuous or continuous depending on the exponents (a, s). For discontinuous phase transitions, a critical shattering intensity λc exists which separates the jammed state from the equilibrium state. For continuous phase transitions, two critical points λc and λc2 separate the jammed state, super-cluster state (a new state) and equilibrium state. The same qualitative behaviour occur in addition with complete/incomplete disintegration with rates Ak = ka and Rk = λkr. Quantitative differences appear in the magnitude of the critical points λc,1 and λc,2 in continuous phase transitions and ?c in discontinuous phase transitions.