## Citation

Zhou, Chengzhe
(2020)
*Collection of Solved Nonlinear Problems for Remote Shaping and Patterning of Liquid Structures on Flat and Curved Substrates by Electric and Thermal Fields.*
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/PEJ5-1626.
https://resolver.caltech.edu/CaltechTHESIS:12092019-191651654

## Abstract

There has been significant interest during the past decade in developing methods for remote manipulation and shaping of soft matter such as polymer melts or liquid metals to pattern films at the micro- and nanoscale. The appeal of low-cost fabrication of micro-optical devices for beam shaping or metallic films to produce high order cuspidal arrays for antireflective or self-cleaning coatings has driven considerable interest in the fundamentals associated with film shaping and liquid curvature. Physicists and applied mathematicians have uncovered new rich ground in examining the complex behavior of high order, nonlinear partial differential equations describing the motion and response of liquid structures driven to redistribute and reorganize by externally applied thermal and electric fields. For the problems relevant to this thesis, which focuses on liquid structures at small scales, the applied fields induce surface forces which act only at the moving interface. Because the surface-to-volume ratios tend to be very large however, the corresponding forces are considerable in magnitude and dominate the formation and growth processes described. In all cases examined, once the driving forces are removed and the operating temperatures dropped below the melting point, the patterned films and liquid shapes rapidly solidify in place, leaving behind structures with molecularly smooth surfaces, an especially advantageous feature for micro-optical applications.

The first part of this thesis examines the nonlinear dynamics of free surface films in the long wavelength limit coating either a flat or curved substrate. We examine the long wavelength limit in which inertial forces are suppressed in comparison to viscous forces such that the system reacts instantaneously to interfacial forces acting in the direction normal to the moving interface, such as capillary and Maxwell forces, or in the direction parallel to the moving interface, such as thermocapillary forces. In the first example, we demonstrate by analytic and numerical means how a system designed to incur large runaway thermocapillary forces can pattern films with conic cusps whose tips undergo self-focused sharpening through a novel self-similar process. This finding expands the known categories of flows that can generate cusp-like shapes and introduces a new lithographic method for remote, one-step fabrication of cuspidal microarrays. We next examine a lithographic technique known as Electrohydrodynamic Lithography in which remote patterned electric field distributions projected onto the surface of a dielectric film generate Maxwell stresses which cause growth and accumulation toward regions of highest field gradients. Here we solve the inverse problem associated with the governing fourth-order nonlinear interface equation by appealing to a control-theoretical approach. This approach reveals the optimal electrode topography required to generate precise complex liquid patterns within a given time interval. Numerical implementation of this algorithm yields high fidelity pattern replication by essentially incorporating proximity corrections which quench undesirable interference effects of material waves. We then extend the long wavelength analysis to a liquid layer coating a curved manifold and demonstrate how a desired film shape can be obtained by novel application of the Helmholtz minimum dissipation principle. We illustrate this solution method by deriving the nonlocal tensorial partial differential equation for the evolution of a slender, perfectly conducting or insulating liquid film supported on a curved electrode. Finite element simulations demonstrate the complex shapes which can result, including formation of liquid accumulation sites and flow instabilities not accessible to films supported on a planar substrate.

The second part of this thesis focuses exclusively on geometric singularities which result from nonlinear effects caused by the coupling of capillary and Maxwell forces in perfectly conducting liquids. Here, we do not restrict ourselves to the long wavelength approximation but instead examine systems with comparable lateral and transverse dimensions. We probe the energy stability of such systems using a convective Lagrangian approach. The exact variational characterization of equilibrium shapes and their stability is derived in the most general form without restriction to coordinate system or shape deformations. This formulation unmasks several terms, typically not evident in calculations restricted to normal deformations of an electrified spherical drop. Our result provides new insights into the energy stability of equilibrium shapes that do not necessarily have constant interface curvature or uniform surface charge distribution. We then turn attention to the classical problem of a conical meniscus produced in an electrified liquid body. The analysis by G. I. Taylor (1963) first determined that the hydrostatic equilibrium shape for a liquid body subject only to capillary and Maxwell forces is given by a cone with an opening angle of 98.6°. However, the fact that such a cone represents an unsteady configuration is often ignored. We revisit the inviscid analysis by Zubarev (2001) who proposed that conic cusps in perfectly conductive liquid evolve through a time-dependent self-similar process. Using the unsteady Bernoulli's equation, he analyzed the force balance at the moving interface and obtained an asymptotically correct self-similar solution dominated by a sink flow far from the evolving apex whose streamlines orient nearly parallel to the moving surface. In addition to the sink flow our analysis, supported by accurate, high resolution numerical solutions of the boundary integral equations, independently reveals a two-parameter family of non-spherically symmetric self-similar solutions whose velocity streamlines intercept the conic surface at an angle. This new family of solutions not only properly account for the interplay between capillary, Maxwell and inertial forces but generate advancing and recoiling type interface shapes, which substantially alter current understanding of the formation and acceleration of dynamic cones.

Item Type: | Thesis (Dissertation (Ph.D.)) | ||||||
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Subject Keywords: | fluid dynamics; nonlinear partial differential equation; optimal control theory; inverse problem; electrohydrodynamics; Riemannian geometry; free surface flow; potential flow; boundary integral equation | ||||||

Degree Grantor: | California Institute of Technology | ||||||

Division: | Physics, Mathematics and Astronomy | ||||||

Major Option: | Physics | ||||||

Awards: | Richard Bruce Chapman Memorial Award, 2020. | ||||||

Thesis Availability: | Public (worldwide access) | ||||||

Research Advisor(s): | - Troian, Sandra M.
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Group: | Laboratory of Interfacial and Small Scale Transport | ||||||

Thesis Committee: | - Bruno, Oscar P. (chair)
- Politzer, Hugh David
- Porter, Frank C.
- Troian, Sandra M.
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Defense Date: | 29 August 2019 | ||||||

Non-Caltech Author Email: | chengzhe.zhou (AT) outlook.com | ||||||

Record Number: | CaltechTHESIS:12092019-191651654 | ||||||

Persistent URL: | https://resolver.caltech.edu/CaltechTHESIS:12092019-191651654 | ||||||

DOI: | 10.7907/PEJ5-1626 | ||||||

Related URLs: |
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ORCID: |
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Default Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||

ID Code: | 13602 | ||||||

Collection: | CaltechTHESIS | ||||||

Deposited By: | Dr. Chengzhe Zhou | ||||||

Deposited On: | 16 Dec 2019 19:31 | ||||||

Last Modified: | 09 Aug 2022 16:07 |

## Thesis Files

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