A test-tube model for rainfall

If the temperature of a cell containing two partially miscible liquids is changed very slowly, so that the miscibility is decreased, microscopic droplets nucleate, grow and migrate to the interface due to their buoyancy. The system may show an approximately periodic variation of the turbidity of the mixture, as the mean droplet size fluctuates. These precipitation events are analogous to rainfall from warm clouds. This paper considers a theoretical model for these experiments. After nucleation the initial growth is by Ostwald ripening, followed by a finite-time runaway growth of droplet sizes due to larger droplets sweeping up smaller ones. The model predicts that the period $\Delta t$ and the temperature sweep rate $\xi$ are related by $\Delta t\sim C \xi^{-3/7}$, and is in good agreement with experiments. The coefficient $C$ has a power-law divergence approaching the critical point of the miscibility transition: $C\sim (T-T_{\rm c})^{-\eta}$, and the critical exponent $\eta$ is determined.