Electrowetting of a 3d Drop: Numerical Modelling with Electrostatic Vector Fields

The electrowetting process is commonly used to handle very small amounts of liquid on a solid surface. This process can be modelled mathematically with the help of the shape optimization theory. However, solving numerically the resulting shape optimization problem is a very complex issue, even for reduced models that occur in simplified geometries. Recently, the second author obtained convincing results in the 2D axisymmetric case. In this paper, we propose and analyze a method that is suitable for the full 3D case. Mathematics Subject Classification. 65N12, 65N30, 49Q10. Received March 26, 2009. Revised October 16, 2009. Published online February 23, 2010. Introduction Handling very small amounts of liquid on a solid surface is of industrial interest, especially in microfluidics. In this field, electrowetting process is now broadly used. Indeed, the typical experiment consists in charging a droplet posed on a solid by applying a given voltage between this droplet and a counter-electrode placed beneath the insulator (see Fig. 1). This allows one to quickly control the wetting of the drop on the solid at small costs. Thus a wide range of applications already exists: from variable focal liquid lenses (Varioptic; Berge, Philips...), to “lab on a chip” and microchip (see e.g.: http://www-leti.cea.fr; Fouillet, http:www. cem.ucla.edu/; Garrell’s group...). The literature on this topic is developed as well; most of it is devoted to experimental aspects. When one lets the applied voltage vary, experiments show a locking phenomenon: after a critical value of the voltage has been reached, the shape of the drop remains fixed. Explaining this locking phenomenon is still an open question: this could be due to droplet ejection, insulator breakdown, ionization of the air (see for example [25,26,31,32]...). Concerning modelling, a first approximate model is commonly used. Called the plane capacitor approximation, it amounts to consider that the system is a plane capacitor. Thus one can describe the behavior of the drop at a given applied voltage. This model has been validated by experiments at small voltages. But it predicts