3D modeling and simulation of the triple junction line movement in Czochralski crystal growth

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The quality of single crystals produced by the Czochralski (Cz) process strongly depends on the shape of the phase change interface, where the crystallization takes place. This shape is influenced by numerous effects such as convection, buoyancy, surface tension, turbulence etc. The most critical issue is the position of the triple phase junction, where the melt, the crystal and the surrounding atmosphere meet, because it controls the interface movement and thus the diameter and resulting shape of the crystal. However, in state-of-the-art 3D numerical simulations this triple junction is never computed explicitly but always locally fixed. This quasisteady-state assumption does not meet the reality.In this project, a method is developed for the numerical description of the movement of the triple junction line. A mathematical relation between the free surface/ meniscus shape and interface shape based on a Laplace-Young equation with an iteratively determined contact angle and a Stefan condition will be utilized. The phase interface as well as the free surface of the melt and thus the triple junction is allowed to move freely, so that fully transient computations of the entire Cz process can be conducted. The moving grid method is used to track the interfaces; it will be improved and extended by elliptic grid smoothing methods to ensure robustness and speed up convergence. Thus it allows to perform time-dependent simulations for many different Cz configurations with various material properties and boundary conditions. Parametric studies are planned conducted to identify the influences relevant for controlling the crystal quality.Remark:The project is conducted in cooperation between the Institute of Fluid Mechanics at the Friedrich-Alexander-University Erlangen-Nürnberg (FAU-LSTM) and the Department of Process Engineering at the University of Applied Sciences Nürnberg (FHN-VT). Through the coupling of fundamental research at the FAU-LSTM with the numerous industrial partners of the FHN-VT, the concerns of small companies will be taken into consideration during the project. Namely the company Siltronic AG, Burghausen/Freiberg, shows deep interest in the present project.