Electric Field-Induced Patterning Theory

Theoretically, the origin of the film instability is best understood by considering the balance of forces acting at the polymer film-air interface. As discussed before, the film surface tension y minimizes the area of the polymer-air interface and stabilizes the film. The electric field on the other hand polarizes the polymer resulting in an effective surface charge density. This result in an electrostatic pressure at the liquid-air interface. An expression for p i is obtained by minimization of the energy, Tel stored in the capacitor with a constant applied voltage, U, as [29]  

The total capacitance, C, is given in terms of a series of two capacitances, the film and the air gap, with dielectric constants 8p and Eair ( 1) respectively. For two capacitors in series (1/C = 1/Cp + 1/CAir), 1/C is given as 

The electrostatic pressure (pel) that destabilizes the film is opposed by the stabilizing Laplace pressure originating from surface tension. The total electrostatic pressure in the film is given by  

The dispersion relation for a system dominated by electrostatic forces is given by [29, 243]  

This is in contrast to the inviscid, gravity stabilized case with q [279] the q variation is a signature of a typical dissipative system [280]. The necessary condition for the amplification of the fluctuations is x 0. Since SiPel 0 ah modes with q = fc = (—0iPel/y) are unstable and are spontaneously amplified. The wavelength of the fastest growing mode (k), which eventually dominate the instability, is given as  

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