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

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$F$isthetotalfreeenergyofthesystemand$t$ is time. There is a conjecture that the long-time average value of $\beta$ should not exceed $1\mathrm{/}3$ – this result is summarized here as $⟨\beta ⟩\le 1\mathrm{/}3$. 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 $⟨\beta ⟩\le 1\mathrm{/}3$. The results for the Cahn–Hilliard equation in the case of symmetric mixtures show $⟨\beta ⟩$ sometimes exceeds $1\mathrm{/}3$ in our simulations. However, the possibility is left open for the very long-time average values of $\langle \beta\rangle$tobeboundedaboveby$1/3$. 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}