The coalescence and bouncing of water drops at an air/water interface

A detailed study has been made of the conditions under which uncharged water drops of radius 60 to 200 $\mu m$ coalesce or rebound at a clean water/air interface. The variable parameters in the system are the drop radius, r, its impact velocity, V$_i$, and the angle of impact, $\theta_i$; and the dependent parameters are the time of contact, $\tau$, between a rebounding drop and the water surface, the velocity, V$_b$, and the angle $\theta_b$ with which it leaves the surface. All these have been measured. Relations are established between the drop radius and the critical values of V$_i$ and $\theta_i$ at which coalescence occurs between uncharged drops and plane or convex water surfaces. Drops impacting at nearly normal incidence remain in contact with the surface for about 1 ms, lose about 95% of their kinetic energy during impact, and rebound with an effective coefficient of restitution of about 0.2. Drops carrying a net charge and drops polarized in an applied electric field coalesce more readily than uncharged drops of the same size and impact velocity. The magnitudes of the critical charges and critical fields required to cause coalescence are determined as functions of V$_i$, $\theta_i$ and drop radius. Typically, drops of radius 150 $\mu m$ impacting at 100 cm/s coalesce if the charge exceeds about 10$^{-4}$ e.s.u. or if the field exceeds about 100 V/cm. If the motion of a drop rebounding from a plane water surface is treated as simple harmonic and undamped, one may derive expressions for the depth of the crater, x, and the restoring force, F, at any stage, and also for the time of contact. These yield values that are in reasonable accord with experiment. However, the collision is clearly inelastic, and a second solution is obtained when F is assumed to be proportional, not only to the displacement, x, but to x/t. This leads to a slightly different expression for the time of contact and to a calculated energy loss of 84% compared with the measured value of 95%. If the drop is to coalesce with the water surface, it must first expel and rupture the intervening air film. Treating the undersurface of the drop as a flattened circular disk, an expression is determined for the minimum thickness, $\delta$, achieved by the film during the period of contact, in terms of V$_i$, $\theta_i$ and the drop radius r. This predicts values of $\delta \sim$0.1 $\mu m$ below which fusion may well take place under the influence of van der Waals forces. Several features of the observed relations between V$_i$, $\theta_i$ and r are accounted for by this simplified theory, but the behaviour of drops impacting at nearly glancing incidence, and of relatively large, energetic drops impacting nearly normally is not. In the latter case, the observed distortion of the drop is thought to play an important role in permitting more rapid thinning of the air film and, in the case of charged and polarized drops, by producing intense local electric fields that may cause the final rupture.