Thin Film Patterning: an Electrostatics-to-Hydrodynamics Inverse Problem and Its Solution

Micro- and nanopatterning techniques for applications ranging from optoelectronics to biofluidics have multiplied in number over the past decade to include adaptations of mature technologies as well as novel lithographic techniques based on periodic spatial modulation of surface stresses. In this talk we focus on shape deformation of nanofilms under the presence of a patterned counter-electrode. Induced polarization charges at the liquid interface causes a patterned electrostatic pressure counterbalanced by surface tension which leads to 3D protrusions whose evolution can be terminated as needed. We formulate a nonlinear evolution equation of liquid film shape governing the electrohydrodynamic response under a preset counter-electrode pattern from variational principle. The corresponding inverse problem of finding designs of counter-electrode pattern that drive film into desired 3D structures is solved by determining optimal control for the nonlinear dynamic system. Optimality conditions are derived and an efficient numerical algorithm is presented. We demonstrate such implementation of film control to achieve periodic, free surface shapes ranging from simple arrays of Gaussian bumps to more complex sawtooth and ring patterns.