Controlling transport phenomena in the Czochralski crystal growth process

A mathematical formulation and computational techniques are presented to describe optimal control and design strategies for the suppression of turbulent motions in the melt and the minimization of temperature gradients in the crystal in Czochralski crystal growth processes. The methodologies developed can be used to test control mechanisms, design parameters, and optimization objectives to determine their effectiveness in improving the processes. They can also be used to effect such improvements by systematically determining optimal values of the design parameters. The controls or design parameters considered include applied magnetic fields, temperature gradients along the side wall of the crucible, and crucible and crystal rotation rates. The results show that applied magnetic fields can be very effective in reducing velocity perturbations in the melt, while side wall temperature gradients are less effective and crucible and crystal rotation rates are ineffective. The results also show that applied magnetic field and crucible and rotation rates are ineffective in reducing temperature gradients in the crystal or in the melt.