Towards a final explanation of the contact angle saturation in electrowetting based on the exploration of the Maxwell stress tensor at a triple point
Fedor Schreiber, Allgemeine und Theoretische Elektrotechnik (ATE), Universität Duisburg-Essen and CENIDE – Center for Nanointegration Duisburg-Essen, D-47048 Duisburg, GermanyAndreas Rennings, Allgemeine und Theoretische Elektrotechnik (ATE), Universität Duisburg-Essen and CENIDE – Center for Nanointegration Duisburg-Essen, D-47048 Duisburg, GermanyDaniel Erni, Allgemeine und Theoretische Elektrotechnik (ATE), Universität Duisburg-Essen and CENIDE – Center for Nanointegration Duisburg-Essen, D-47048 Duisburg, Germany
We present a rigorous analysis of the (maximal) electric force at a conductive wedge formed by the wetting angle of a liquid droplet on an insulated electrode using the Maxwell stress tensor yielding a final explanation for the so-called contact angle saturation in electrowetting. When applying a voltage on the liquid droplet placed on a insulated electrode, the resulting forces will be confined to the tip of the wedge – represented by a triple junction of the three adjacent phases: conductive liquid, dielectric and surrounding air.
These confined forces tend to drag the droplet over the hydrophobic dielectric surface (electrowetting effect) causing a deformed liquid surface with a decreased contact angle. Forces acting solely on the triple point while being described by corresponding surface tensions together with an interfacial shear force define the realm of the approximate Young-Lippmann formalism, where the voltage dependent contact angle becomes the major parameter for the emergent drag force. It is though not surprising that the contact angle saturation at a minimal angle (yielding maximal drag force) is subject to a still unresolved debate relating its origin to various microscopic but rather disconnected mechanisms.
Here, we present a macroscopic explanation of the contact angle saturation that has the potential to predominate the microscopic ones due to emergent electrostatic field singularities in the triple point. In our theoretical 2D analysis, we calculated the electric field and flux density in a close radial neighbourhood around the triple point relying on their well-known fractional order local dependence. Using the Maxwell stress tensor together with renormalization techniques, it is shown that the estimated electrostatic «pressure» (force/m²) in the singularity together with the adjacent much weaker ones are strongly dependent on the contact angle, where the horizontal component of this resulting pressure changes direction (with a very steep zero crossing) at a contact angle of 80°. Below this angle any droplet transport is inhibited and thus any further decrease of the contact angle.