Mathematical modelling of dye-sensitized solar cells

Dye-sensitized solar cells remain a prevalent object of study in electrochemistry since their inception in 1991. Their potential to provide low cost renewable energy has motivated decades of research in pursuit of suitable materials to maximise performance and efficiency. In this thesis, we give an account of the primary components found in dye-sensitized solar cells to inform modelling approaches. Mathematical modelling has great potential to determine optimal materials and configurations of dye-sensitized solar cells for maximising efficiency or reducing production costs. In this thesis we give an account of the models commonly used in the literature. In particular, we find that throughout the literature, diffusion-based approaches focussed on electron transport governing dye-sensitized solar cells allow for evaluation of the current-voltage relationship. Given diffusion is the main mechanism driving electron transport across dye-sensitized solar cells, this thesis focuses on a diffusion-based model given by a time dependent boundary value problem. We analyse each component of a dye-sensitized solar cell in order to derive mathematical models capable of capturing its influence on electron transport. In particular, the nanoporous nature of the semiconductor is a crucial component to electron transport. To account for the trap states in a given nanoporous semiconductor, we consider a natural extension of the diffusion model to include nonlinear diffusivities. Finally, based on continuous-time random walk simulations of titanium dioxide, commonly used as a nanoporous semiconductor, we develop a new diffusion equation utilising the Caputo fractional derivative. To solve these boundary value problems the literature generally employs explicit finite difference methods. In this thesis we use Lie symmetry analysis to explore the possibility of obtaining analytical solutions for nonlinear diffusion models. For the cases which do not admit analytical solutions we provide alternative numerical solutions based on Runge-Kutta, finite element and cubic B-Spline collocation methods in addition to implicit finite difference methods. These numerical schemes offer superior stability and accuracy by comparison to explicit finite difference methods, especially for the nonlinear diffusion equation and the fractional diffusion equation. Using the solution of the electron density from the diffusion equation, we calculate the current-voltage relationship for a given dye-sensitized solar cell to ultimately determine its efficiency.