Scaling law for the critical voltage of a droplet on a surface in the presence of an external field

Using external fields to regulate the droplet shape and behavior has attracted considerable attention due to its wide range of practical applications. However, determining the complex flow phenomenon inside a droplet and its deformable boundary shape at equilibrium is a challenging physical and mathematical exercise. In the present work, we employ theoretical and experimental approaches to study the shape deformation of a sessile droplet on a substrate in the presence of an external electric field. Based on the theoretical model we propose, and by combining the finite element method with the gradient descent algorithm, we successfully determine the droplet shape by minimizing the total free energy of the system, viz., a combination of electrostatic energy, surface tension energy, and gravitational potential energy. We also perform scaling analyses and derive an empirical expression for the critical voltage, featuring a universal scaling exponent of 1/2 for the contact angle as a function of normalized volume. The master curve depicted by the empirical expression provides an excellent fit to both the experimental and numerical results.