Electro-Energy-Reaction-Diffusion Systems

Reaction and diffusion processes appear in numerous areas of everyday life. To give only a few examples, chemical reactions are the basis for photosynthesis, combustion, and the operation of batteries, while the physical phenomena of drift and diffusion are fundamental for the propagation of heat, groundwater, and polluted air. Electrically charged particles, which also attract and repel each other, occur for example in batteries, semiconductors, and electrolysis appliances. In most cases, there is also a subtle interplay with the temperature. On the one hand, the temperature generally changes in the course of the above processes but, on the other hand, the temperature itself influences the rate at which these processes take place. Our starting point is a general model (more precisely, a system of equations), which takes the above mentioned effects into account. For given initial density profiles of the involved species, for example electrons and ions in the case of semiconductors, these equations determine the temporal evolution of the densities. As our first goal, we seek to prove that space- and time-dependent density profiles exist which solve our system of equations for all times. The fact that such a global solution exists for all times is far from being obvious, and even on a finite interval of time, the rigorous construction of a solution in this context poses various challenges. Another part of our project is concerned with a specific state-of-the-art model arising in the theory of organic semiconductors. While the majority of classical semiconductors is built up of silicon, organic semiconductors generally consist of carbon-based chemical compounds. As the electronic properties of organic semiconductors typically show a pronounced temperature dependence, it is essential to also take the temperature into account. Proving that a global solution for the density profiles of the charge carriers exists for all times is the objective of this study. As soon as a global solution is known to exist, one can investigate its evolution for large times. For a system modeling chemical reactions in a classical way, we expect that the density profiles of the involved species approach limiting densities, called the equilibrium, as time tends to infinity. The equilibrium is a special solution of the model in the sense that it remains constant as time evolves. Finally, we aim to show that the rate at which the global solution approaches the equilibrium can be explicitly estimated in terms of given model parameters.

Understanding the mechanisms of real-world reaction-diffusion processes taking temperature and electrostatic effects into account remains a challenging task. On the one hand, the corresponding mathematical models are highly complex due to the coupling between the different phenomena. On the other hand, it is not fully settled how to combine the different effects into a unified framework. For instance, we had to update a specific aspect of our originally proposed model in order to be consistent with fundamental physics, even though this aspect is found in the standard literature. Our first result is concerned with equilibrium states. These are special solutions to our model, which do not change as time evolves, and which maximize the total entropy under the constraints of charge and energy conservation. We proved that a unique equilibrium exists by providing two proofs highlighting different viewpoints of the maximization problem. Moreover, we proved that all (time-dependent and sufficiently regular) solutions to a special class of electro-energy-reaction-diffusion systems approach the equilibrium as time tends to infinity. This class contains, in particular, a fundamental model for electron-hole recombination in inorganic semiconductors. We could even show that the solutions, i.e., the concentrations of electrons and holes, approach the equilibrium at a minimal rate, for which an explicit formula in terms of model parameters can be given. Open problems, which could not be settled within this project, are the construction of so-called global solutions for general electro-energy-reaction-diffusion systems and for general organic semiconductor models. Global solutions can be seen as time- and space-dependent concentrations of, e.g., electrons and holes, which solve the corresponding system of equations for all times. In the former case, it is mainly the electrostatic interaction, which interferes with available strategies for energy-reaction-diffusion systems. In the latter case, the main obstacle is the absence of upper and lower bounds on the temperature, which are crucial for constructing global solutions.