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 $\beta=-(t/F)(d F/d t)$, 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 $\langle \beta\rangle\leq 1/3$. 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 $\langle\beta\rangle\leq 1/3$. The results for the Cahn--Hilliard equation in the case of symmetric mixtures show $\langle\beta\rangle$ sometimes exceeds $1/3$ in our simulations. However, the possibility is left open for the very long-time average values of $\langle \beta\rangle$ 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}