Exchange Driven Growth

TL;DR

This paper analyzes a class of exchange-driven growth processes, identifying regimes of indefinite growth, gelation, and instant gelation, with detailed distribution behaviors and applications to physical systems.

Contribution

It introduces a comprehensive theory for exchange-driven growth, characterizing different regimes and their distribution properties, and applies it to physical models like the Ising-Kawasaki system.

Findings

Clusters can grow indefinitely or form a gel depending on exchange rates.

The size distribution exhibits self-similar form with specific tail behaviors.

Gelation time scales logarithmically with system size.

Abstract

We study a class of growth processes in which clusters evolve via exchange of particles. We show that depending on the rate of exchange there are three possibilities: I) Growth: Clusters grow indefinitely; II) Gelation: All mass is transformed into an infinite gel in a finite time; and III) Instant Gelation. In regimes I and II, the cluster size distribution attains a self-similar form. The large size tail of the scaling distribution is Phi(x) ~ exp(-x^{2-\nu}), where nu is a homogeneity degree of the rate of exchange. At the borderline case nu=2, the distribution exhibits a generic algebraic tail, Phi(x)\sim x^{-5}. In regime III, the gel nucleates immediately and consumes the entire system. For finite systems, the gelation time vanishes logarithmically, T\sim [\ln N]^{-(\nu-2)}, in the large system size limit N\to\infty. The theory is applied to coarsening in the infinite range Ising-Kawasaki model and in electrostatically driven granular layers.