Critical charges for droplet collisions

Two micron-sized water droplets approaching each other do not always coalesce due to the cushioning effect of the air between them. When the droplets do not carry any electrical charges, one needs to consider the breakdown of hydrodynamics at very small scales to decide whether the droplets collide and coalesce or not. In contrast, two approaching droplets that are oppositely charged always coalesce if the charges are large enough. To find the charge for which the transition to charge-dominated collisions occurs, we computed the collision efficiency of charged, hydrodynamically interacting droplets settling in quiescent air, including the noncontinuum regime at small interfacial distances. For oppositely charged droplets, we find that the transition occurs when a saddle point of the relative droplet dynamics exits the region within which the continuum hydrodynamics breaks down. For cloud droplets with radii 16 and $20\mu \mathrm{m}$, we observe that the transition occurs at $\sim {10}^3$ elementary charges $e$. For charges smaller than this, we predict that the coalescence rate depends primarily upon the Knudsen number ($\mathrm{Kn}$, the ratio of the mean-free-path of air to the mean droplet radius), whereas coalescence for much larger charges does not depend upon $\mathrm{Kn}$. For droplets charged with the same polarity, we find the critical charge to be substantially larger ($\sim {10}^{4}e$ for the above radii) for reasons that we discuss.