Non-self-similar behavior in the LSW theory of Ostwald ripening

TL;DR

This paper investigates the long-time behavior of particle size distributions in Ostwald ripening, showing that classical self-similar solutions are not always approached, depending on initial conditions.

Contribution

It demonstrates that the classical self-similar behavior predicted by LSW theory does not hold universally, providing criteria for convergence to alternative solutions.

Findings

Convergence to classical similarity solutions is impossible for certain initial distributions.

The initial distribution's behavior near the support boundary determines long-term dynamics.

For many initial conditions, the system does not approach any self-similar solution.

Abstract

The classical Lifshitz-Slyozov-Wagner theory of domain coarsening predicts asymptotically self-similar behavior for the size distribution of a dilute system of particles that evolve by diffusional mass transfer with a common mean field. Here we consider the long-time behavior of measure-valued solutions for systems in which particle size is uniformly bounded, i.e., for initial measures of compact support. We prove that the long-time behavior of the size distribution depends sensitively on the initial distribution of the largest particles in the system. Convergence to the classically predicted smooth similarity solution is impossible if the initial distribution function is comparable to any finite power of distance to the end of the support. We give a necessary criterion for convergence to other self-similar solutions, and conditional stability theorems for some such solutions. For a dense set of initial data, convergence to any self-similar solution is impossible.