## Doctoral Dissertations

#### Date of Award

12-2017

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy

#### Major

Mathematics

#### Major Professor

Steven Wise

#### Committee Members

Ohannes Karakashian, Tim Schulze, Michael Berry, Rajeev Kumar

#### Abstract

We consider in this dissertation the mathematical modeling and simulation of a general diffuse interface mixture model based on the principles of energy dissipation. The model developed allows for a thermodynamically consistent description of systems with an arbitrary number of different components, each of which having perhaps differing densities. We also provide a mathematical description of processes which may allow components to source or sink into other components in a mass conserving, energy dissipating way, with the motivation of applying this model to phase transformation. Also included in the modeling is a unique set of thermodynamically consistent boundary conditions which allows flow across the boundary of a select number of components. The result of this modeling is a unique Cahn-Hilliard, Allen-Cahn-like system of equations. For numerical solution of this model, we use cell-centered finite difference methods for discretization and Full Approximation Storage (FAS) multigrid methods to solve the resulting system of equations via use of the BSAM (Block- Structured Adaptive Multigrid) libraries. Upon development of the mathematical model, we consider two applications.

The primary application of this mathematical modeling is the time evolution of a quaternary mixture consisting of a volatile solvent in the liquid phase, solvent in the vapor phase, and two polymers. This modeling is motivated by the need to better understand the active layer in Organic Photovoltaics (OPVs). In this mixture, the volatile solvent is evaporating into the its vapor phase, and upon fully evaporating the polymer mixture which results is the active layer of the OPV device. Simulations are provided which demonstrate the solvent evaporation phenomenon and the resulting microstructure of the active layer.

As a future application, we consider a mixture of a charged polymer and its counterion. We provide a description of the system based on the dissipation of the electrochemical free energy which allows for the permittivity to be dependent on the volume fractions. Simulations are provided which vary the gradient energies and polymer chain length and demonstrate the different steady-state microstructures which can result.

#### Recommended Citation

Cummings, John Timothy, "Mathematical Modeling of Mixtures and Numerical Solution with Applications to Polymer Physics. " PhD diss., University of Tennessee, 2017.

https://trace.tennessee.edu/utk_graddiss/4739

#### Included in

Numerical Analysis and Computation Commons, Other Applied Mathematics Commons, Partial Differential Equations Commons