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

Czochralski(Cz)法生产的单晶的质量很大程度上取决于晶化发生的相变界面的形状。这一形状受到对流、浮力、表面张力、湍流等多种效应的影响。最关键的问题是三相结的位置，即熔体、晶体和周围大气相交的位置，因为它控制着界面的运动，从而控制晶体的直径和最终的形状。然而，在最先进的3D数值模拟中，这种三重连接从来不是显式计算的，而总是局部固定的。这种准稳态假设与实际情况不符。在本项目中，提出了一种数值描述三重结合线运动的方法。将利用基于具有迭代确定的接触角的Laplace-Young方程和Stefan条件的自由面/弯月面形状和界面形状之间的数学关系。允许熔体的相界面和自由面自由移动，从而允许三重结自由移动，从而可以进行整个Cz过程的全瞬变计算。界面跟踪采用移动网格法，通过椭圆网格法对其进行改进和扩展，以保证算法的稳健性和收敛速度。因此，它允许对具有不同材料属性和边界条件的许多不同的Cz构型进行随时间变化的模拟。计划进行参数研究，以确定与控制晶体质量相关的影响。备注：该项目是由弗里德里希-亚历山大-纽伦堡大学流体力学研究所(FAU-LSTM)和纽伦堡应用科学大学过程工程系(FHN-VT)合作进行的。通过将FAU-LSTM的基础研究与FHN-VT的众多工业合作伙伴相结合，小公司的关切将在项目期间得到考虑。也就是说，世创电子材料公司博格豪森/弗莱贝格对本项目表现出了浓厚的兴趣。

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.