Optimal Terminal Control of the Thin Film Equation for Electrohydrodynamic Patterning of a Dielectric Film

The projection of a spatially periodic electric field pattern onto the surface of a slender dielectric liquid film can be used to fabricate microlens and other micro-optical arrays by a process known as electrohydrodynamic lithography. Prior to solidification, the formation and nonlinear growth of these periodic arrays is governed by the spatiotemporal competition between electric and capillary forces acting at the moving interface. Theoretical work has mostly focused on solution and identification of various regimes related to the forward problem wherein the thin film equation is solved numerically subject to different initial and boundary conditions. However, since the governing equation is fourth order and highly nonlinear, it is not possible to intuit or identify exactly which electric field distribution pattern will generate a particular microarray shape within a specified time interval. Here we discuss results of a terminal control strategy based on the thin film equation for achieving specific curved shapes in 1D and 2D relevant to micro-optical systems. The results delineate those parameter regimes leading to high fidelity patterning as a function of the target shape geometry, electric Weber number and termination time.