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Charged Induced Viscous Fingering in Toroidal Droplets

Alexandros Fragkopoulos, Aaron Aizenman, Alberto Fernandez-Nieves

Spherical droplets are stable. Surface tension tends to minimize the surface area for a given volume and as a result a sphere does not spontaneously change shape. In order to cause a constant motion of the interface of a sphere there must be some kind of external field. Even by placing a droplet in an electirc field, it deforms into a spheroid, but still does not exhibit a spontaneous motion. In previous studies, we have found that by changing th etopology of a droplets, we can cause it to spontaneously exhibit motion.

Video. 1: A torus shrinking before breaking.

A toroidal droplet ehxibits a shrinking behavior, where the handle of the torus shrinks, as shown in Video 1. The reason of this evolution is due to the Laplace pressure: $$ p_{in}-p_{out} =2\gamma H $$ which describes the pressure difference across a liquid-liquid interface due to surface tension. In this equation, \(p_{in}\) and \(p_{out}\) are the pressures of the inner and outer liquids at the interace, respectively, \(\gamma\) the surface tension, and \(H\) the mean curvature. A torus does not have constant mean curvature. Specifically, the pressure is higher on the outer circumference of the torus than inside, and this pressure difference causes the torus to shrink. As a result the torus moves in a quasi 2-dimensional plane.

Video. 2: A charged torus expands before it breaks.

The addition of charge has been explored in a different project (Electrically Charged Toroidal Droplets). On eof the key features of this research was that charge can qualitatively change the behavior of a torus. Specifically, the torus does not shrink anymore, but it rather expands due to the electrostatic stresses. This is due to the change of the normal stress balance at the interface: $$ p_{in}-p_{out} =2\gamma H -\frac{1}{2}\epsilon E^2$$ where \(\epsilon\) the dielectric constant of the outer liquid, and \(E\) the electric field at the interface. The negative sign shows that this stress is in competition with the capillary stresses, and for high enought voltage the torus expnads (see video). Interestingly, if we lower the surface tension, we find another qualitatively different feature, that is that the torus produces finger like structures as it evolves (see video).

Video. 3: A charged torus exhibiting fingering instability.

In this research project, we study this intersting phenomenology that arises from the topology of the droplet. We try to understand the reason behind this change, as well as the features of the fingers, for example, number of fingers, and the width.

[1] D. R. Nelson, Nano Lett. 2, 1125 (2002).
[2] T. C. Lubensky and J. Prost., J. Phys. II (France) 2, 371 (1992).

Soft Condensed Matter Laboratory, School of Physics, Georgia Institute of Technology
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alberto.fernandez [at] physics.gatech.edu