caa a22 4500 475797396 CHVBK 20180406123728.0 cr unu---uuuuu 170329e20000301xx s 000 0 eng 10.1007/s000330050198 doi (NATIONALLICENCE)springer-10.1007/s000330050198 Inertial effects in the rotationally driven melt motion during the Czochralski growth of silicon crystals with a strong axial magnetic field [Elektronische Daten] [G. Talmage, S.-H. Shyu, J. S. Walker, J. M. Lopez] Abstract.: This article treats the melt motion driven by the rotations of the crystal and crucible about their common vertical axis during the Czochralski growth of silicon crystals with a strong, uniform, vertical magnetic field produced by a solenoid around the crystal growth furnace. Since molten silicon is an excellent electrical conductor, the interaction parameter N and the Hartmann number Ha are both large for typical magnetic field strengths, so that composite singular perturbation techniques for $N\gg1$ and $Ha\gg1$ are appropriate. An inertialess solution, which also assumed that $N\gg Ha^{3/2}$ , was presented in a previous paper. In the inertialess solution, the largest gradient of the azimuthal velocity $v_\theta$ and the largest secondary flow with radial and axial velocities v r and v z both occur inside an interior layer with an $O(Ha^{-1/2})$ radial thickness at the vertical cylinder directly beneath the periphery of the crystal, where the growth interface meets the free surface. For all current experimental studies, the assumption that $N\gg Ha^{3/2}$ is not satisfied. The appropriate assumption is that $N=O(Ha^{3/2})$ , and inertial effects are not negligible inside the interior layer. An intersection region, which is formed by the intersection of the interior layer and a Hartmann layer with an $O(Ha^{-1})$ axial thickness adjacent to the crystal-melt interface and free surface, is intrinsically coupled to the interior layer. This article treats inertial effects in the interior layer and intersection region for $N=O(Ha^{3/2})$ and $Ha\gg1$ . Non-linear governing equations were derived and solved numerically. A fourth-order Adams-Bashforth-Moulton predictor-corrector method was used to solve the transport equations for the primary azimuthal velocity and for the secondary-flow vorticity. Poisson equations, which govern the stream functions for both the secondary flow and the electric current density, were solved using a matrix diagonalization technique. The effects of inertia on the melt motion are discussed. This type of study provides for a fuller understanding of the melt motion, without which defect-free crystals will be difficult to grow on a consistent basis. Birkhäuser Verlag, Basel,, 2000 Key words. Magnetic Czochralski crystal growth, inertial effects, rotationally driven flow, asymptotic analysis nationallicence Talmage G. Dept. of Mech. & Nucl. Eng., The Pennsylvania State University, University Park, Pennsylvania 16802, U.S.A., e-mail: gxt3@psu.edu, US aut Shyu S.-H Dept. of Mech. & Nucl. Eng., The Pennsylvania State University, University Park, Pennsylvania 16802, U.S.A., e-mail: gxt3@psu.edu, US aut Walker J. S. Dept. of Mech. & Indust. Eng., The University of Illinois at Urbana, Urbana, Illinois 61801, U.S.A., US aut Lopez J. M. Department of Mathematics, Arizona State University, Tempe, Arizona 85287, U.S.A., US aut https://doi.org/10.1007/s000330050198 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s000330050198 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Talmage G. Dept. of Mech. & Nucl. Eng., The Pennsylvania State University, University Park, Pennsylvania 16802, U.S.A., e-mail: gxt3@psu.edu, US aut NATIONALLICENCE 700 1- Shyu S.-H Dept. of Mech. & Nucl. Eng., The Pennsylvania State University, University Park, Pennsylvania 16802, U.S.A., e-mail: gxt3@psu.edu, US aut NATIONALLICENCE 700 1- Walker J. S. Dept. of Mech. & Indust. Eng., The University of Illinois at Urbana, Urbana, Illinois 61801, U.S.A., US aut NATIONALLICENCE 700 1- Lopez J. M. Department of Mathematics, Arizona State University, Tempe, Arizona 85287, U.S.A., US aut Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer