In this thesis, we compute approximate solutions to initial value problems of first-order linear ODEs using five explicit Runge-Kutta methods, namely the forward Euler method, Heun's method, RK4, RK5, and RK8. This thesis aims to compare the accuracy, stability, and efficiency properties of the five explicit Runge-Kutta methods. For accuracy, we carry out a convergence study to verify the convergence rate of the five explicit Runge-Kutta methods for solving a first-order linear ODE. For stability, we analyze the stability of the five explicit Runge-Kutta methods for solving a linear test equation. For efficiency, we carry out an efficiency study to compare the efficiency of the five explicit Runge-Kutta methods for solving a system of first-order linear ODEs, which is the main focus of this thesis. This system of first-order linear ODEs is a semi-discretization of a two-dimensional wave equation.

The aim of this thesis is to modify and improve the n-type semiconducting polymer PNDIT2 for thermoelectric generators (TEGs) applications. Although the PNDIT2 is considered a prime n-type material due to its high electron mobility, low air-stability of radical anions after doping and the low doping efficiency with molecular dopants are severe drawbacks and lead to limited application in TEGs. To this end, the backbone structure of PNDIT2 is modified by polymer analogous thionation and the branched aliphatic side chains are replaced by branched, fully oligoethylene glycol-based side chains. PNDIT2 was prepared by DAP and subjected to various conditions of thionation. The polymer analogous thionation of PNDIT2 was done by using Lawesson´s reagent (LR). The O/S conversion was controlled by solvent, T and amount of LR. For an excess of LR, only two carbonyls out of four present in the NDI repeating unit are converted to thiocarbonyls with regioselective trans-conformation (2S-trans-PNDIT2). Chlorobenzene (CB) is an excellent solvent in which the highest O/S conversion was achieved and the best reproducibility. Tri- or tetra- substitution in one NDI repeat unit did not take place due to steric hinderance of T2 comonomer. Thionation affected all properties. The lower thermal stability, UV-vis spectra were bathochromically shifted and a new band of the thionated NDI unit appeared. Chain aggregation was stronger as probed by UV-vis and NMR spectroscopy. The LUMO energy level of 2S-trans-PNDIT2 was lowered by 0.2 eV, giving -4.0 eV. This is at the border of what is needed for air stability of radical anions. The scattering on thin films indicated lower order and less crystalline textures of 2S-trans-PNDIT2 compared to PNDIT2. Likewise, electron mobility decreased with increasing conversion. While chapter 2 focused on the synthesis, opto-electronic and thermal properties of 2S-trans-PNDIT2, chapter 3 was concerned with details on morphology and electrical properties. To this end, 2S-trans-PNDIT2 was doped by N-DPBI in toluene at various concentrations and conductivities were determined. Undoped 2S-trans-PNDIT2 exhibited one order of magnitude higher conductivity than pristine PNDIT2. After doping with 5 wt.-% N-DPBI, the conductivity of 2S-trans-PNDIT2 increased by two orders of magnitude and reached a maximum conductivity of 6*10-3 S/cm at 15 wt.-% doping. This value was approx.5 times higher than the conductivity of PNDIT2 at the same doping level. Furthermore, the stability of conductivity of doped 2S-trans-PNDIT2 under ambient conditions was investigated and compared to PNDIT2. Upon exposure air (50 % humidity), conductivity of PNDIT2 rapidly decreased to the pristine film level, while the conductivity of 2S-trans-PNDIT2 was reduced by a factor of less than two after 16 h. While the initially higher conductivity of 2S-trans-PNDIT2 is ascribed to its less crystalline structure and thus higher doping efficacy, its better stability can be ascribed to the lower LUMO energy level. The topic of chapter 4 is on the synthesis of fully ether-based, polar and branched side chains (EO) and introduction into PNDIT2. The advantages of polar side chains over aliphatic side chains have been reported. However, previously reported PNDIT2 with linear polar side chains is limited in MW due to solubility. The EO side chain with amine functionality was synthesized in three steps and used for monomer synthesis (EO-NDIBr2). Initial efforts to use DAP to prepare P(EO-NDIT2) from EO-NDIBr2 and pristine bithiophene gave only oligomeric products. Stille polycondensation was therefore used, giving high MW. As extreme aggregation occurred in solvents used for GPC, absolute MW were determined by 1H NMR spectroscopy. To enable reliable end group analysis, model compounds with methyl end groups were prepared. In P(EO-NDIT2), methyl end groups are dominating as a result of incorrect transmetalation from the stannylated monomer. The end groups seen by 1H NMR spectroscopy were further confirmed by MALDI-ToF. Absolute MW were between Mn,NMR= 11 kg/mol to 116 kg/mol depending on reaction conditions. Aggregation was further probed by UV-vis and NMR spectroscopy as a function of the solvent and temperature, shining light into the degree of aggregation, which is important for thin film preparation. Solvent quality decreased with the following order: CHCl3, 1-Chloronaphthalene (CN), 1,2-Dichlorobenzene (o-DCB), DMF, 1,4-Dioxane, CB and Anisole (AN). According to these results, three doping protocols based on CB and o-DCB, as well as temperature variations, were used to prepare films for conductivity measurements. The best results were obtained for processing from chlorobenzene at 80 °C, which aggregates are dissolved. Strikingly, maximum conductivity values were achieved already for 5 wt.-% dopant concentration. The PF reached a maximum even for 1 wt.-% doping level. This unusually low value is promising and suggests a high doping efficacy.