A second-order turbulence simulation of the Czochralski crystal growth melt: the buoyantly driven flow

Abstract

The statistics of the fluctuating velocity and temperature fields in a large scale Czochralski silicon crystal grower are simulated using a Reynolds stress turbulence closure. The methodology and some of the ideas underlying such a simulation method are discussed. Simulations of the buoyancy driven case are presented: for Grashof numbers Gr = 109 and 1010, and Marangoni numbers Ma = 2 × 103, 15 × 103,and60 × 103. The solution procedures, which averages over small scale fluctuations, indicates unsteady mean flow measure with time scales on the order of the integral time scale. The source of the flows unsteadiness is the occurrence, growth and dissipation of eddies between the crucible wall and the free surface. This time scale corresponds to time scales seen in the dopant striations. The simulations indicate two very active areas in the melt flow: at the outer edge of the crystal-melt interface and at the bottom center of the crucible. Underneath the free surface the temperature fluctuations are comparatively small. The material and structural inhomogeneities in the grown crystal are related to the variance and skewness of the temperature fluctuations and the vertical turbulent heat flux at the crystal melt interface.