The European Physical Journal Special Topics

, Volume 149, Issue 1, pp 1–17 | Cite as

Computation of nonlinear multiscale coupling effects in liquid phase epitaxy

  • V. Chalupecký
  • Ch. Eck
  • H. Emmerich


A new two-scale model for liquid phase epitaxy is presented which enables the numerical simulation of processes with microstructures having an arbitrarily small scale. It is based on a BCF-model for epitaxial growth, a Navier–Stokes system and a convection-diffusion equation. The application of a homo- genization approach leads to a separation of scales; the resulting two-scale model consists of macroscopic partial differential equations for fluid flow and solute diffusion in the fluid volume, coupled to microscopic BCF-models. The two-scale model can be discretized using grids that are independent of the scale of the microstructure. Numerical experiments based on a phase field version of the BCF model are presented; the results illustrate the physical relevance of the model.


European Physical Journal Special Topic Epitaxial Layer Liquid Phase Epitaxy Dislocation Cell Lead Order Term 
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  1. E. Bauser, Handbook of Crystal Growth, Vol. 3, edited by D.T.J. Hurle (North-Holland, Amsterdam, 1994) Google Scholar
  2. W.K. Burton, N. Cabrera, F.C. Frank, Phil. Trans. Roy. Soc. 243, 299 (1951) Google Scholar
  3. G. Caginalp, Arch. Ration. Mech. Anal. 92, 205 (1986) Google Scholar
  4. A.A. Chernov, T. Nishinaga, Growth Shapes and Stability in Morphology of Crystals, edited by I. Sungawa (Terra Scientific Publ. Co., 1987), p. 270 Google Scholar
  5. W. Dorsch, S. Christiansen, M. Albrecht, P.O. Hansson, E. Bauser, H.P. Strunk, Surf. Sci. 896, 331 (1994) Google Scholar
  6. Ch. Eck, in Multiscale Modeling and Simulation, Vol. 3 (2004), p. 28 Google Scholar
  7. Ch. Eck, H. Emmerich, Models for Liquid Phase Epitaxy, Preprint 146 – DFG SPP 1095 “Mehrskalen probleme” (2004) Google Scholar
  8. H. Emmerich, Ch. Eck, Cont. Mech. Thermodyn. 17, 373 (2006) Google Scholar
  9. R. Ghez, Int. J. Heat Mass Transfer 23, 425 (1980) Google Scholar
  10. L.D. Khutoryanskii, P.P. Petrov, Sov. Phys. Crystallogr. 23, 571 (1978) Google Scholar
  11. F.P.J. Kuijpers, G.F.M. Beenker, J. Cryst. Growth 48, 411 (1979) Google Scholar
  12. M.B. Small, E. Ghez, E. Gies, in Handbook of Crystal Growth, Vol. 3, edited by D.T.J. Hurle (North-Holland, Amsterdam, 1994) Google Scholar
  13. V.V Jikov, S.M. Kozlov, O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals (Springer, Berlin, 1994) Google Scholar
  14. H. Müller-Krumbhaar, J. Chem. Phys. 63, 5131 (1975) Google Scholar
  15. W.W. Mullins, R.F. Sekerka, J. Appl. Phys. 35, 444 (1964) Google Scholar
  16. E.M. Sparrow, J.L. Gregg, Trans. ASME J. Heat Transfer 82C, 294 (1960) Google Scholar
  17. N. Tokuda, J. Cryst. Growth 67, 358 (1984) Google Scholar
  18. W.R. Wilcox, J. Cryst. Growth 12, 93 (1972) Google Scholar
  19. L.O. Wilson, N.L. Schryer, J. Fluid Mech. 85, 479 (1978) Google Scholar

Copyright information

© EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2007

Authors and Affiliations

  • V. Chalupecký
    • 1
  • Ch. Eck
    • 2
  • H. Emmerich
    • 3
  1. 1.Department of MathematicsFaculty of Nuclear Science and Physical Engineering, Czech Technical University in PraguePragueCzech Republic
  2. 2.Institute for Applied Mathematics, Friedrich-Alexander UniversityErlangenGermany
  3. 3.Computational Materials Engineering, Center for Computational Engineering Science, Institute of Minerals Engineering, RWTH Aachen UniversityAachenGermany

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