An Observer Design for Temperature Estimation in Czochralski Crystal Growth Process With Time-Varying Domain

Czochralski crystal growth process is a well-known process used for single crystal production. The crystal growth starts with a small seed crystal and growth happens at the melt-crystal interface while drawing crystal from a pool of the melted crystal material [1] Due to high-tech applications of single crystals, crystal quality is the most important characteristics that should be considered in the growth process. Temperature profile in the crystal has an important role in defects concentration and residual stresses inside the crystal [2]. In order to monitor and control the temperature, knowledge of temperature over the entire domain is necessary, despite the fact that temperature measurements on the entire domain are not available and cannot be measured directly. In order to monitor the temperature distribution in crystal, an estimation strategy is required to estimate crystal temperature and construct temperature profile over the entire domain. The temperature evolution dynamic is expressed by a parabolic partial differential equation on a moving boundary domain. The domain geometry is determined by the radius and length evolution modeled by a hydro-geometrical dynamics. Time-varying domain results in a time-varying PDE on a time-varying domain. In this work, an ODE model of pulling dynamics is coupled with 2-D temperature evolution dynamics expressed by time-varying PDE on a moving boundary. Using the Galerkin's model the 2-D parabolic PDE is approximated and reduced to a low dimensional ODE considering the time-varying domain effects . Observability of the system is investigated and an observer is synthesized based on point boundary temperature measurements. Finally, the observer is implemented on the finite element model (FEM) of the crystal and the efficiency of the temperature estimation and effects of uncertainties in modelling is studied. The crystal geometry and temperature distribution are shown in figure below at different times during the process.

References

[1] M. A. Gevelber, D. Wilson and N. Duanmu, Modeling requirements for development of an advanced Czochralski control system, Journal of Crystal growth, 230 (2001) 217-223

[2] M. A. Gevelber, G. Stephanopoulos, M. J. Wargo, Dynamics and Control of the Czochralski Process II. Objectives and control structure design, Journal of Crystal growth, 91 (1984) 199-217