Studies of the Evolution and Stability of the Thin Film Equation for Externally Modulated Control of Electrohydrodynamic and Thermocapillary Patterning

Citation

Chang, Yi Hua (2024) Studies of the Evolution and Stability of the Thin Film Equation for Externally Modulated Control of Electrohydrodynamic and Thermocapillary Patterning. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/t65x-a578. https://resolver.caltech.edu/CaltechTHESIS:05092024-225430822

Abstract

It has been known for a couple decades, based on extensive experimental, theoretical and numerical studies, that a flat slender nanoscale viscous film in the absence of gravity always undergoes early time linear instability when subject to electrical or thermocapillary forces. The patterns resulting from a uniform transverse electric or thermal field resemble clusters of small rounded protrusions whose early time dynamics have been described by linear stability analysis of the governing fourth-order nonlinear interface equation -- the so-called thin film equation. However, the pattern formation process beyond early times generates larger amplitude protrusions prone to coalescence or an Oswald-like ripening of adjacent formations which destroy the pattern uniformity. Introduction of film interface modulation by external spatially periodic modulation offers a superior method for this type of lithographic patterning. The resulting linear and nonlinear response of the liquid layer can be tuned to corral the evolution of the liquid interface into periodic arrays containing identical components in certain parameter range. Conditions for achieving high-fidelity patterns are still not fully understood, however, rendering such technique not yet fully utilized in practical applications.

To that end, we have conducted a number of analytical and numerical studies which elucidate various regimes leading to high-fidelity patterning by external spatial and temporal modulation. We focus on a single layer of viscous liquid film on a solid substrate which is described by the thin film equation derived under the long wavelength approximation. We first study the linear stability of periodic non-uniform stationary states subject to electrostatic stress and find that the necessary conditions for achieving stable states in 1D are the mass-limitation or saturation with a system-confining boundary (touching the mask) in order to suppress the coalescence and Ostwald-like-ripening modes. In 2D, stationary ridges are only achieved by saturation with a system-confining boundary in order to suppress its breakup. Time-dependent simulations further reveal inaccessible stationary states due to large electrode separation or large applied voltage. Exploratory studies on system subject to temperature gradient shows that the coalescence mode becomes unstable over a wider range of parameters due to thermocapillary stress. These findings result in phase diagrams relating the spatial modulation amplitude and electric Weber number or Marangoni number to the conditions for high-fidelity patterns which cannot be explained simply by matching the patterning and intrinsic instability wavelengths as previously claimed in literature.

We then turn to the optimal control of electrohydrodynamic thin film patterning where the optimal strategy in deforming a flat film toward a desired shape is determined. A computational framework is derived which allows us to study the open-loop terminal control problem for thin liquid film. The approach allows us to quantify the best-possible outcome only constrained by the underlying physical mechanisms, and better understand the limitations of thin film patterning in relation to the choice of target shapes and system parameters. The impact of imperfect engineering and methods of mitigations are also discussed, which should prove useful to soft lithography and other applications.

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