PhD Dissertation
Mathematical Models in Microfluidics:
Capillary Electrophoresis & Sessile Drop Physics
by Yousef Daneshbod and Ali Nadim (Advisor).
Abstract
In this thesis two different microfluidic applications are modeled mathematically and solved by analytical and numerical methods. These include (i) modeling kinetics of binding interactions during electrophoresis, and (ii) determining the static shapes of sessile droplets on various patterned surfaces and subject to electrostatic actuation. These problems cover the two most important approaches in the emerging field of microfluidics, which are micro-channel-based (or continuous) and micro-droplet based (or digital).
For the micro-channel case, we focus upon on-chip capillary electrophoresis of multiple interacting species. Specifically, electrophoretic transport of three chemically reacting species, two of which can bind reversibly to form a third, is mathematically modeled. The species are assumed to move horizontally through a long channel with different electrophoretic mobilities and diffusion coefficients.
By considering small perturbations of the system about equilibrium, or when one of the two binding species is much more abundant than the other, the governing advection-reaction-diffusion equations can be linearized and studied via the method of moments. The result is a set of coupled ordinary differential equations for the moments that can be solved analytically. Analysis of the longtime evolution of the moments yields mean velocities and dispersion coefficients for each species. The results provide a method for measuring the rate and equilibrium constants of binding reactions using capillary electrophoresis.
For the case of digital or droplet microfluidics, we focus on determining the shape of sessile drops on patterned surfaces or under electrowetting actuation. The current motivation for studying this problem is the fact that the development and optimization of fluidic devices require a detailed understanding of interfacial phenomena and the influence of energy modulations. In this dissertation we investigate both analytically and numerically the surface morphology of liquids deposited onto lithographically patterned surfaces and influenced by electrical fields. In our approach, the equilibrium shape of a constant volume droplet on a patterned surface is determined by minimizing the total free energy, which includes all contributions from body, surface and electrostatic forces. A novel formulation, in the form of a relaxation-type partial differential equation with constraints, is presented along with a numerical procedure, based on finite differences, to solve for a multitude of drop topologies.
Ph.D. Thesis, School of Mathematical Sciences, Claremont Graduate University
July 7, 2006
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