Equilibrium shapes and stability of a liquid film subjected to a nonuniform electric field

Abstract

Stresses induced by a spatially nonuniform electric field acting on an initially flat fluid-fluid interface can (i) be exploited beneficially to pattern polymer microstructures without the use of resists, exposure, development, and etching, but (ii) cause undesirable nonuniformity in film thickness in precision coating processes. The equilibrium shape of an interface separating a liquid film from an ambient fluid subjected to a uniform electric field is flat so long as the field strength is below a critical value. A nonuniform electric field, however, results in the deformation of the interface no matter how small its strength, an important difference which previous theoretical studies have not addressed satisfactorily. Hence, whereas under a uniform field loss of stability occurs via a bifurcation from the flat film solution, under a nonuniform field destabilization may occur at a turning point at which the film profile already exhibits a finite-amplitude deformation. This deficiency in understanding is remedied here by analyzing a model problem in which a gas overlying a perfect dielectric liquid film is sandwiched between two electrodes wherein the top electrode is grounded and the electric potential of the bottom electrode varies sinusoidally with distance measured along it. The equilibrium shapes and stability of the liquid-gas interface are determined directly in the present work by simultaneously solving the augmented Young-Laplace equation governing the shape of the free surface and the Laplace equation governing electric potentials theoretically by domain perturbation analysis and numerically by finite element analysis. For small nonuniformities in the electric field, analytical solutions are reported for the profile of the free surface. The computational predictions are shown to be in excellent accord with these small-deformation results. Moreover, computations are used to extend the investigations into the nonlinear regime where nonuniformities in the electric field and deformations of the free surface are large, and loss of stability may occur. The variation of the equilibrium shapes and the limits of stability with the governing dimensionless groups are investigated thoroughly. It is shown that the rich response exhibited by the system can be rationalized by interrogating the computed solutions and scrutinizing the balance of stresses due to the normal component of the electric field, which are destabilizing, and those due to its tangential component, which are stabilizing.