View source for 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 <math>\Epsilon_c</math>. For sufficiently high values of <math>\Epsilon_c</math>, 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 ==

The experimental setup consists of two needles separated by a gap d. A power generator supplies a controllable voltage <math>\beta</math> across the needles. Drops are then formed using deionized water. Once the drops reach a critical size, they deform and are brought together by the electric force. The authors observe the drop dynamics using a camera, and they look at many different sizes and voltages. They also conduct experiments in different gases in order to test for effects caused by the dielectric strength of the surrounding gas.

The experimental setup is shown in Fig 1a. The authors observe two distinct behaviors. At low voltages the drops form a growing fluid neck that leads to coalescence (Fig 1b). At high voltages the drops deform, come into contact, but then repel and recoil (Fig 1c). After they recoil they approach a second time and coalesce (perhaps because they are no longer conical). All of this occurs within a millisecond. The authors are interested in the recoil dynamics after the first contact. 

[[Image:bird_prl1.jpg|400px|thumb|center|Fig. 1. (a) A voltage <math>\Delta \phi</math> 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 <math>\beta</math>. At lower voltages, the drops immediately coalesce; <math>\Delta \phi</math> = 815 V. (c) At higher voltages, the drops contact and then recoil;  <math>\Delta \phi</math>  = 822 V. Note the slight misalignment of the needles in (a) demonstrates the robustness of this phenomenon.]]

The cone angle is positively correlated with applied voltage and negatively correlated with the needle separation (Fig 2). For a given separation distance, there is a critical voltage when the drops no longer coalesce (Fig 2). 

How to explain the recoil at high voltages? The authors present 3 possible reasons and reject the first 2. First, high voltage could cause dielectric breakdown in a gas. But if this were the case, an increase in the dielectric strength of the gas would require a larger critical voltage to bring about the transition. This is contradicted by experiment (Fig 2 inset). Secondly, the recoil could be a consequence of Joule heating. But Joule heating is proportional to conductivity, and the critical voltage does not change when the fluid conductivity is varied (Fig. 2).

[[Image:bird_prl2.jpg|500px|thumb|center|Fig. 2. The cone angle <math>\beta</math> 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, <math>\sigma</math> = 4 <math>\mu</math>S/cm, d = 2.4 mm (up triangles), and <math>\sigma</math> = 163 <math>\mu</math>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. ]]

Finally, the authors hypothesize that the coalescence-recoil transition is a consequence of drop geometry. When two spherical drops touch, the contact generates capillary waves, which create an expanding liquid neck
between the drops. The negative curvature across the neck is much larger than the positive curvature around the neck, leading to lower fluid pressure in the neck than in the center of the drop. This capillary pressure difference drives fluid into the meniscus resulting in drop coalescence (Fig 3a). In contrast, the instability in a liquid thread leads to a neck region with a negative curvature across the neck that is much smaller than the positive curvature around the neck (Fig 3b): the capillary pressure in the neck is positive, creating a flow away from the region and eventually causing the thread to break up. These physical
arguments suggest that a coalescence-breakup transition may occur for conical drops, as the geometry typically has comparable curvatures across and around the neck (Fig 3c). 

[[Image:bird_prl3.jpg|600px|thumb|center|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.]]

The authors then do some modeling and find a critical angle of 30.8 degrees, which is in good agreement with the experimental transition in Fig. 2. In this model, the elctric field sets the cone angle; after contact, the dynamics depend on capillary effects. This explains why the transition is independent of fluid conductivity, and why the data in Fig. 2 collapses onto a master curve when rescaled by <math>\Epsilon_c</math>.

== Discussion ==

== Relation to Soft Matter ==