The treatment of convective boundaries remains an important source of uncertainty within stellar evolution, with drastic implications for the thermally-pulsing stars on the Asymptotic Giant Branch (AGB). Various sources are taken as motivation for the incorporation of convective boundary mixing during this phase, from s-process nucleosynthesis to hydrodynamical models. In spite of the considerable evidence in favour of the existence of convective boundary mixing on the pre-AGB evolution, this mixing is not universally included in models of TP-AGB stars. The aim of this investigation is to ascertain the extent of convective boundary mixing, which is compatible with observations when considering full evolutionary models. Additionally, we investigate a theoretical argument that has been made that momentum-driven overshooting at the base of the pulse-driven convection zone should be negligible. We show that, while the argument holds, it would similarly limit mixing from the base of the convective envelope. On the other hand, estimations based on the picture of turbulent entrainment suggest that mixing is possible at both convective boundaries. We demonstrate that additional mixing at convective boundaries during core-burning phases prior to the thermally-pulsing Asymptotic Giant Branch has an impact on the later evolution, changing the mass range at which the third dredge-up and hot-bottom burning occur, and thus also the final surface composition. In addition, an effort has been made to constrain the efficiency of convective boundary mixing at the different convective boundaries, using observational constraints. Our study suggests a strong tension between different constraints that makes it impossible to reproduce all observables simultaneously within the framework of an exponentially decaying overshooting. This result calls for a reassessment of both the models of convective boundary mixing and the observational constraints.

Current implementations of the stellar atmosphere typically derive boundary conditions for the interior model from either grey plane-parallel atmospheres or scaled solar atmospheres, neither of which can be considered to have appropriate underlying assumptions for the Thermally Pulsing Asymptotic Giant Branch (TP-AGB). This paper discusses the treatment and influence of the outer boundary condition within stellar evolution codes, and the resulting effects on the AGB evolution. The complex interaction of processes, such as the third dredge up and mass loss, governing the TP-AGB can be affected by varying the treatment of this boundary condition. Presented here are the results from altering the geometry, opacities and the implementation of a grid of MARCS/COMARCS model atmospheres in order to improve this treatment. Although there are changes in the TP-AGB evolution, observable quantities, such as the final core mass, are not significantly altered as a result of the change of atmospheric treatment. During the course of the investigation, a previously unseen phenomena in the AGB models was observed and further investigated. This is believed to be physical, although arising from specific conditions which make its presence unlikely. If it were present in stars, this phenomenon would increase the carbon-star lifetime above 10Myr and increase the final core mass by ∼ 0.1M in the narrow initial-mass range where it was observed (∼ 2 − 2.3M ).