# Capillary and electrostatic limitations to the contact angle in electrowetting-on-dielectric

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## Abstract

The shape of a conducting liquid droplet placed on a hydrophobic dielectric surface is simulated numerically by solving the Laplace–Young capillary equation. The electric force, acting on the conducting surface, distorts the droplet shape leading to a change in the apparent contact angle; its variation is compared with a theoretical Young–Lippman prediction. At sufficiently large values of voltage, applied to the droplet, the numerical algorithm fails to converge, which is interpreted as the break-up of the droplet surface with small droplets being ejected from the surface. These highly charged droplets, as well as any other electric charges near the triple contact line, generated for example by the electric corona discharge, cause a change of the distribution of the electric forces. This effect can be helpful in explaining saturation of the apparent contact angle: an appropriately selected surface charge near the contact line can completely stop droplet distortion, and the contact angle variation, despite the increased droplet voltage. Furthermore, the simulation results show the effect of the permittivity of the medium surrounding the droplet, on the contact angle variation.

## Keywords

Electrowetting Electrocapillary effect Contact angle Droplets Electric field## List of symbols

- 1/
*R*_{1}*+*1/*R*_{2} sum of the principal radii of curvature

*d*thickness of the dielectric layer

*g*gravity constant

**n**unit vector normal to the surface (versor)

*r*radial coordinate in the spherical set of coordinates

*f*_{φ}derivative of function

*f*with respect of*φ***D**electrostatic displacement vector

**E**electric field intensity

*R*radius of an undistorted droplet

*V*scalar electric potential

*V*_{0}voltage applied to the droplet

*ε*,*ε*_{m}absolute electrical permittivity of the solid insulating layer and ambient medium

*ε*_{r}, \(\varepsilon _{{{\text{rm}} }}\)relative electrical permittivity of the solid insulating layer and ambient medium

*γ*surface tension

*θ*apparent contact angle (also called Lippman’s angle)

*θ*_{0}local contact angle (Young’s angle)

*ρ*fluid density

*σ*_{s}surface charge density

*σ*space charge density

*φ*azimuthal coordinate in the spherical coordinate system

- Δ
*p* pressure difference between any pair of points inside and outside of the droplet

## Indices

- s
solid

- l
liquid

- n
normal direction

## Notes

### Acknowledgment

The work was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC).

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