Date of Award

5-2023

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematical Sciences

First Advisor

Jian Du

Second Advisor

Yi Liao

Third Advisor

Vladislav Bukshtynov

Fourth Advisor

Xianqi Li

Abstract

Ion-induced volume phase transitions in polyelectrolyte gels play an important role in physiological processes such as mucus storage and secretion in the gut, nerve tissue excitation, and DNA packaging. Biological experiments show that polyelectrolyte gels may swell or collapse rapidly due to changes in external conditions such as ionic composition. The volume phase transition is accompanied by a monovalent/ divalent ion exchange between the polymer network and the solvent that make up the gel. We propose a 2D computational method for simulating mucus swelling and deswelling with a two-fluid mixture model. The model includes electro-diffusive transport of ionic species, the coupled motion of the polymer network and hydrating fluid, and chemical interactions between the network and dissolved ions. Each ionic species in the solvent phase is subject to a Nernst–Planck type equation. Together with the electro-neutrality constraint, these equations constitute a system of non-linear parabolic partial differ

ential equations subject to an algebraic constraint. The discretized system is solved by a novel Schur complement reduction scheme. Numerical results indicate that the method is very efficient, robust and accurate, even for problems which exhibit large spatial gradients in the concentration of ions. The computational effectiveness of the new methods is demonstrated through accuracy and efficiency metrics and through investigation of some of the factors that influence ion-induced volume phase transitions. The chemical forces acting on the gel are responsible for the unique volume phase transitions that occur within the gel. They include entropic, short-range, and electric forces, which explicitly depend on the bath environment as well as the chemical and structural properties of the gel. A major discovery from our experimental investigations is that depending on the charge distribution on the network, as well as the advective and diffusive transport of the ionic species, the induced electric field may either promote or oppose the swelling of a polyelectrolyte gel. Another significant discovery is that different components of the short range interaction force can drive swelling or deswelling of the gel. Depending on the properties of the gel, different components of the short-range force can have a competing effect on the swelling or deswelling behavior, yielding unique pattern formation within the polyelectrolyte gel. Finally, this work introduces and develops a novel computational framework for simulating multiphase mixture models on an adaptively refined mesh. The adaptive mesh refinement (AMR) technique is necessary for simulating complicated fluid models in order to gain speed enhancements and improved accuracy for features that require high resolution. Our preliminary results demonstrate the capabilities of the AMR fluid solver in adaptively refining regions of interest, thereby reducing the computational cost when simulating the coupled motion of the polymer network and solvent.

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