Electric-field-induced patterns in thin polymer films: Weakly nonlinear and fully nonlinear evolution

A thin polymer melt on a substrate can be unstable to an electric field normal to the interface, a phenomenon that can be harnessed as a patterning technique with a range of potential applications. Motivated by the variety of patterns observed in experiments for polymers under both unpatterned and patterned masks, we describe here, from theoretical and numerical analyses, how nonlinear effects govern the growth of the instability and determine the final patterns. In particular, we discuss the nonlinear growth in terms of interactions among different Fourier modes and show that the second- and third-order nonlinearities favor the growth of hexagonal patterns under a featureless mask, in agreement with experimental observations. Also, numerical simulations based on the fully nonlinear model validate the prediction of the weakly nonlinear analysis: hexagonal patterns do emerge under an unpatterned mask. Furthermore, in one-dimensional simulations, we demonstrate the energetic evolution of this patterning process and reveal several kinetically stable structures along the path to the thermodynamically stable state. Two-dimensional simulations allow us to study the effects of both mask patterns and the initial film thickness. Generally, patterns on the mask guide the growth such that the pattern conforms to the geometric shapes. Interestingly, a small cylindrical protrusion at the center of the mask can produce exactly the same pattern as a large, flat, circular protrusion. The initial film thickness or the thickness ratio of the polymer layer to the air gap plays an important role in determining the final pattern formed. Finally, we demonstrate, by two simple examples, that the simulations can provide insights on smart mask designs for producing large areas of well-ordered patterns.