Numerische Simulation von Erstarrungsprozessen : Wege zur mehrskaligen Modellierung ausgehend von Mikrostrukturmodellen

The field where this thesis can be ranged in is the modeling of solidification processes. A central problem in this modeling is the large range of length scales on which important processes take place during a solidification. The comprehensive consideration of all or at least many of these scales in a single model is currently accomplished only to some extent. However, the multi-scale modeling is indispensable to take a step from a qualitative towards a quantitative modeling, especially for application-orientated solidification scenarios. In this thesis two multi-scale approaches for the modeling of the single-phase dendritic solidification of a binary alloy system are being studied. The starting point for this is a microscopic sharp-interface model. This model describes in a classical and rigorous way the thermodynamics for a solid-liquid-phase interface driven by diffusion gradients and includes the phase-interface tension as well as kinetic effects and their anisotropies. The first discussed multi-scale approach bases on the separation of the transport scales of the heat conduction and the solutal diffusion. The motivation for this separation is founded on the fact that these two transport scales differ by several orders of magnitude for most alloys. This difference impedes an efficient investigation of the original sharp-interface model without scale separation with a full consideration of the microstructure evolution for a macroscopic section of an alloy melt. However, the two-scale model which results from the separation allows to describe the growth of a periodic array of equiaxed dendritic nuclei which interact through the released latent heat over a macroscopic temperature field. At first, analytical investigations of the two-scale model are discussed in this PhD thesis. In these investigations two growth regimes appear in dependence on the density of the equiaxed nuclei in the considered domain. For a low nuclei density a kineticly governed growth of the nuclei results while for a high nuclei density a capillary growth regime arises. Furthermore, numerical investigations of the two-scale model are discussed. In these investigations the influence of the ratio of the two separated transport scales on the growth dynamics of the microstructures is studied. Here, the existence of two different growth regimes appears in dependence on the value of the scale parameter. However, the underlying mechanisms of these two growth regimes could not be identified yet. The reason for this is the considerable demand of numerical resources for the simulation of the two-scale model despite of the efficiency-increasing scale separation. At this point the need for further deeper investigations arose to fathom the cause of these two growth regimes. The second multi-scale approach discussed in this PhD thesis bases on a phenomenological coupling of the microstructure modeling with a purely macroscopic model for the directional solidification of a cylindrical bottom-cooled sample of a binary alloy. This approach of scale coupling was investigated at the example of the hypoeutectic alloy system Pb-25wt%Sn. Therefore, microstructure simulations were performed with the already mentioned sharp-interface model for columnar dendritic growth within the framework of the PhD thesis. The primary spacings of the columnar dendrites were evaluated in dependence on the cooling rate and the temperature gradient. The comparison with simplified theoretical models from the literature showed that the primary-spacing dynamics under neglect of kinetic effects at the phase interface in the microstructure simulations coincides qualitatively with the predictions of these models. If, however, the kinetic effects are considered in the simulations in the sense of a comprehensive thermodynamical description then the resulting primary-spacing dynamics does not coincide anymore with that from the simplified models from the literature. With the use of the columnar microstructure dynamics from the simulations in the macroscopic solidification model an improved conformance of the spatial structure and temporal evolution of the temperature field with the measurement data from respective experiments manifests. This PhD thesis shall illustrate that it is feasible to include a comprehensive exact microstructure modeling in a multi-scale modeling if solidification processes and that the microstructure modeling is moreover even essential for a more quantitative multi-scale solidification modeling