Critical charges for droplet collisions

Abstract

The collision efficiency of uncharged micron-sized water droplets in air is determined by the breakdown of hydrodynamics at droplet separations of the order of the mean-free path, by van-der-Waals forces, or a combination of the two. In contrast, electrostatic forces determine the collision efficiency of charged droplets if the charge is 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 breakdown of hydrodynamics at small interfacial distances. For oppositely charged droplets, the transition occurs when a saddle point of the relative droplet-dynamics exits the region where the hydrodynamics breaks down. For droplets with radii $16\,μ$m and $20\,μ$m, this occurs at $\sim 10^3$ elementary charges $e$. For smaller charges, the collision efficiency depends upon the Kn number (defined as the ratio of the mean-free-path of air to the mean droplet radius), whereas for larger charges it does not. For droplets charged with the same polarity, the critical charge is $\sim 10^4\,e$ for the above radii.

Figures (2)

• Figure 1: Panel ( a) shows a schematic of two droplets with radii $a_1<a_2$ settle in a quiescent fluid. The separation vector between their centres-of-mass is $\hbox{\boldmath$R$}$. Also shown is the collision sphere around the smaller droplet (dashed). The droplets collide if the centre-of-mass of the larger one hits the collision sphere. Gravity points in the negative $R_3$-direction. Panel ( b) illustrates two regimes of the collision dynamics, distinguished by the location of the fixed point S$1$ (see text). For weak charges, S$1$ is inside the region where hydrodynamics breaks down (hashed area), at non-dimensional distance $s^\ast\equiv R^\ast-2<{\rm Kn}$. For strong charges, the fixed point lies outside, $s^\ast>{\rm Kn}$.
• Figure 2: Collision efficiencies and phase portraits of charged droplets settling in still air. ( a) Collision efficiency $\mathcal{E}$ for droplets with equal amounts of charge of opposite parity, as a function of charge $q=|q_{1,2}|$ over elementary charge $e$. Droplet radii: $a_1 = 16 \,\mu$m and $a_2 = 20 \,\mu$m (blue), and $a_1 = 8 \,\mu$m and $a_2 = 10 \,\mu$m (red). Kn $=10^{-3}$ (solid lines), $10^{-2}$ (dashed), and $5\times10^{-2}$ (dash-dotted). The vertical solid lines indicate the locations where Eqs. \ref{['eq:theory_dim']} and \ref{['eq:details']} predict a transition. Arrows denote the charges in panels ( b, c). Panel ( b) shows the relative droplet dynamics in the $R_1$-$R_3$-plane (where $R_i$ is non-dimensionalised by $\bar{a}$), for $a_1 = 16\,\mu$m, $a_2=20\,\mu$m, $q_{1,2}=\pm 908 \,e$, and Kn $=10^{-3}$. Colliding trajectories (blue), non-colliding trajectories (red), The solid black line is the separatrix between colliding and non-colliding trajectories, and $\bullet$ represents the saddle point S$1$. ( c) Same as ( b), but for $q_{1,2}=\pm 15469\,e$. Manifolds of the saddle point S$1$ ($\bullet$) are shown as black solid lines. ( d) Same as ( a), but for equal parity. Panel ( e) shows the bifurcations (their locations are denoted by vertical dashed lines) that give rise to saddle points S$2$ and S$3$. Panel ( f) shows the relative dynamics in the $R_1$-$R_3$-plane for $a_1 = 16\,\mu$m, $a_2=20\,\mu$m, $q_{1,2}=6011\,e$ along with the saddle point S$1$, and panel ( g) corresponds to $q_{1,2}= 21199 \,e$, and shows the saddle point S$2$.