A finite element method for analysis of fluid flow, heat transfer and free interfaces in Czochralski crystal growth

P. A. Sackinger, R. A. Brown, J. J. Derby

Research output: Contribution to journalArticlepeer-review

80 Scopus citations

Abstract

A finite element algorithm is presented for simultaneous calculation of the steady state, axisymmetric flows and the crystal, melt/crystal and melt/ambient interface shapes in the Czochralski technique for crystal growth from the melt. The analysis is based on mixed Lagrangian finite element approximations to the velocity, temperature and pressure fields and isoparametric approximations to the interface shape. Galerkin's method is used to reduce the problem to a non‐linear algebraic set, which is solved by Newton's method. Sample solutions are reported for the thermophysical properties appropriate for silicon, a low‐Prandtl‐number semiconductor, and for GGG, a high–Prandtl–number oxide material. The algorithm is capable of computing solutions for both materials at realistic values of the Grashof number, and the calculations are convergent with mesh refinement. Flow transitions and interface shapes are calculated as a function of increasing flow intensity and compared for the two material systems. The flow pattern near the melt/gas/crystal tri‐junction has the asymptotic form predicted by an inertialess analysis assuming the meniscus and solidification interfaces are fixed.

Original languageEnglish (US)
Pages (from-to)453-492
Number of pages40
JournalInternational Journal for Numerical Methods in Fluids
Volume9
Issue number4
DOIs
StatePublished - Apr 1989

Keywords

  • Czochralski crystal growth
  • Finite element method
  • Free boundary problem
  • Heat transfer
  • Incompressible fluid flow

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