Dynamic Modeling of Critical Velocities for the Pushing/Engulfment Transition in the Si-SiC System Under Gravity Conditi
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INTRODUCTION
IN the crystallization of silicon for photo-voltaics, the melt conditions determine the formation and distribution of the particles in the solid state. This is evidenced by the experimental studies on SiC and Si3 N4 particles and their distribution in industrial crystalline silicon systems have been reported in recent years[1–4] which provide important information for the photo-voltaic technology of solar cells. The inclusions of SiC and Si3 N4 , if they form under non-optimized growing conditions, can lead to short-circuit currents and have a detrimental effect on the efficiency of the silicon solar cell. A difference in density between SiC particles and the silicon melt can cause sedimentation effects on the crucible bottom. During a directional solidification process, a movement of the phase boundary determines the critical velocity of the solidification front to the particle capture. The morphology of the particle shape as well as the gravity force can also influence the capture of particles in the growing crystal. Therefore, for a detailed study of the installation behavior of these particles and a determination of the critical growth velocity under
microgravity conditions, it is necessary to eliminate gravity-driven effects during crystallization. In the present work, we aim to find out why existing models to particle capture are not able to describe the experimental observations in the laboratory and to develop improved models. The phenomenon of particle pushing by a moving solidification front takes attention of scientists since the 60s. This kind of pushing has been described by Uhlmann et al.[5] as a steady-state process where the particle and the front move with the same velocity, and the gap width between the particle and front remains constant. It was shown that particle pushing occurs only when the front is moving below a certain velocity, known as the critical velocity (Vc ). This velocity is a function of the solidification velocity and some material parameters such as interfacial energies, thermal conductivity. Many experimental as well as numerical models for the calculation of Vc have been developed over the last years.[6–14] The starting point of these models is to consider the physical forces acting on a particle in vicinity of the liquid-solid interface as shown in Figure 1. There are considered:
Interface force, FI, which can be produced by Van der Waals forces.
Lift force, FL, which is caused by the melt flow JULIA KUNDIN and HENNING AUFGEBAUER, Postdoctoral Researchers, are with the Department of Engineering Science, University Bayreuth, 95448, Bayreuth, Germany. Contact e-mail: [email protected] CHRISTIAN REIMANN, Group Manager, JAN SEEBECK, Postdoctoral Researcher, and JOCHEN FRIEDRICH, Head of Department, are with Fraunhofer IISB, 91058, Erlangen, Germany. THOMAS JAUSS and TINA SORGENFREI, Postdoctoral Researchers, and ARNE CROELL, Professor, are with the University of Freiburg, 79104, Freiburg, Germany. Manuscript submitted January 15, 2016. METALLURGICAL AND MATERIAL
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