# Doctoral Dissertation Defense: Preston Donovan

## Advisor: Dr. Muruhan Rathinam

Friday, November 3, 2017

3:15 PM - 5:00 PM

3:15 PM - 5:00 PM

Mathematics/Psychology : 401

**Title:**

*Homogenization Theory for Solute Motion Accounting for Obstructions, Interactions, and Path Length Effects*

**Abstract**

This thesis studies homogenization theory for the prediction of a diffusing particle's motion in an environment containing periodic obstructions. We investigate this problem in the context of a solute diffusing in an aqueous polymer environment, such as polymer solutions and gels. The study of solute diffusion in polymer gels has many important biomedical applications such as drug delivery, separations, and cell encapsulation. We explore a framework based on homogenization theory for the prediction of macro-scale solute diffusivity in an environment containing fine-scale heterogeneities. We assume the solute is subjected to obstructed diffusion via periodic and stationary obstructions. First, we explore the feasibility of this framework using established PDE homogenization theory and then comparing with experimental data.

Our work provides a proof-of-concept for the validity of the homogenization theoretic paradigm for this problem. The solute undergoes specular reflections when interacting with the surrounding obstructions. Via comparison with experimental data, we confirm that homogenization theory accurately predicts the macro-scale effective diffusivity of various solute-polymer pairs satisfying certain assumptions. Next, we consider a periodic, weighted, directed graph and the associated continuous time Markov process $Z(t)$. This discrete setting conveniently accounts for the nonzero path length of the diffusing particle, complex obstruction geometries, and solute-polymer interactions such as attraction, repulsion, and bonding. The process $Z(t)$ takes place on the graph's nodes and jumps along the graph's edges with jump rates given by the edge weights. The jump rates (edge weights) are not assumed to be reversible.

We demonstrate that, under certain conditions on the graph, the scaled process $\eps Z(t/\eps^2)$ converges weakly in the Skorokhod space to a Brownian motion. We show that reversible rates, detailed balance, and certain symmetries of the graph are each alone sufficient for the convergence. For the case of reversible jump rates, an equivalent variational formulation is derived.