A large-scale statistical study of the coarsening rate in models of Ostwald-Ripening

Abstract

In this article we look at the coarsening rate in two standard models of Ostwald Ripening. Specifically, we look at a discrete droplet population model, which in the limit of an infinite droplet population reduces to the classical Lifshitz--Slyozov--Wagner model. We also look at the Cahn--Hilliard equation with constant mobility. We define the coarsening rate as $\beta =-(t/F)(dF/dt)$, where $F$ is the total free energy of the system and $t$ is time. There is a conjecture that the long-time average value of $\beta$ should not exceed $1/3$ -- this result is summarized here as $⟨\beta ⟩\le 1/3$. We explore this conjecture for the two considered models. Using large-scale computational resources (specifically, GPU computing employing thousands of threads), we are able to construct ensembles of simulations and thereby build up a statistical picture of $\beta$. Our results show that the droplet population model and the Cahn--Hilliard equation (asymmetric mixtures) are demonstrably in agreement with $⟨\beta ⟩\le 1/3$. The results for the Cahn--Hilliard equation in the case of symmetric mixtures show $⟨\beta ⟩$ sometimes exceeds $1/3$ in our simulations. However, the possibility is left open for the very long-time average values of $⟨\beta ⟩$ to be bounded above by $1/3$. The theoretical methodology laid out in this paper sets a path for future more intensive computational studies whereby this conjecture can be explored in more depth. \end{abstract}

27 pages, 20 figures

References in corpus (1)

• Bubbles and Filaments: Stirring a Cahn-Hilliard Fluid