# Critical Angle for Electrically Driven Coalescence of Two Conical Droplets

Entry by Leon Furchtgott, APP 225 Fall 2010.

J. C. Bird, W. D. Ristenpart, A. Belmonte, and H. A. Stone, "Critical Angle for Electrically Driven Coalescence of Two Conical Droplets," Physical Review Letters 103 (16) (2009)

## Summary

The authors use a combination of experimental and numerical methods to explore the coalescence behavior of two oppositely-charged conical droplets. They find that coalescence behavior depends on the cone angle: two drops coalesce when the slopes of the cones are small but recoil when the slopes exceed a critical value. They estimate and measure a critical cone angle of 30.8 degrees.

## Background

Two drops of the same liquid are expected to coalesce when they come into contact because the combined drop minimizes the surface energy. When drops fail to coalesce, it is often because of phenomena that prevent contact such as surfactant or colloidal coatings or dynamic processes such as evaporation or vibration that maintain a layer of immiscible fluid between drops. This paper is concerned with the case of two oppositely charged drops placed in large electric fields that fail to coalesce even when the two drops come directly into contact.

Electric fields cause liquid droplets to develop conical structures oriented in the direction of the field. These are known as Taylor cones and result from a balance of charge-induced pressure from the applied electric field and capillary pressure resisting interfacial deformation. The balance of the two effects is quantified by a dimensionless number $\Epsilon_c$. For sufficiently high values of $\Epsilon_c$, two drops develop conical tips and converge.

The question that this paper addresses is the previously unstudied behavior of the drop pairs after contact. They show that for sufficiently high fields, water droplets deform into steep cones and fail to coalesce. The rest of the paper documents this experimentally and gives explanations for this behavior.

## Results

Fig. 1. (a) A voltage $\Delta \phi$ is applied across a pair of drops that are suspended on needles separated by a distance d (here d = 1.9 mm). (b) Prior to contact, each drop deforms into a cone with angle $\beta$. At lower voltages, the drops immediately coalesce; $\Delta \phi$ = 815 V. (c) At higher voltages, the drops contact and then recoil; $\Delta \phi$ = 822 V. Note the slight misalignment of the needles in (a) demonstrates the robustness of this phenomenon.
Fig. 2. The cone angle $\beta$ varies based on the voltage and size of the drops. After contact, the drops either coalesce (open symbols) or recoil (closed symbols). The coalescence-recoil transition for deionized water drops at various needle separations, d = 1.9 mm (circles), 4.3 mm (squares), and 5.1 mm (diamonds), occurs at different voltages, yet at similar cone angles. These features are consistent with previously reported data that focus on varying conductivity, $\sigma$ = 4 $\mu$S/cm, d = 2.4 mm (up triangles), and $\sigma$ = 163 $\mu$S/cm, d = 2.8 mm (down triangles). Inset: The critical voltage between coalescence and recoil decreases, rather than increases, with the dielectric strength of surrounding gas.
Fig. 3. (a) When two spherical drops contact, the curvature of the neck leads to a low local liquid pressure resulting in inward flow and coalescence. (b) The local curvature in an unstable liquid thread leads to a high pressure resulting in outward flow and pinch off. (c) We propose that immediately after two conical drops contact, there is a self-similar neck region, which we approximate as a volume-conserving minimal surface. We then calculate if the resulting curvature raises or lowers the local pressure, which is responsible for the resulting fluid motion.