Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-05T09:44:27.205Z Has data issue: false hasContentIssue false
  • English
  • Français

Pattern Formation in Directional Solidification

Published online by Cambridge University Press:  01 February 2011

Mike Greenwood ,
Mikko Haataja  and
Nikolas Provatas
Mike Greenwood
Affiliation:
McMaster University, Department of Materials Science and Engineering, 1280 Main Street West, Hamilton, Ontario, Canada, L8S 4L7 (Dated: January 26, 2004)
Mikko Haataja
Affiliation:
McMaster University, Department of Materials Science and Engineering, 1280 Main Street West, Hamilton, Ontario, Canada, L8S 4L7 (Dated: January 26, 2004)
Nikolas Provatas
Affiliation:
McMaster University, Department of Materials Science and Engineering, 1280 Main Street West, Hamilton, Ontario, Canada, L8S 4L7 (Dated: January 26, 2004)
Get access

Abstract

We simulate directional solidification using the phase field method solved with adaptive mesh refinement. We examine length scale selection for two cases. For small surface tension anisotropy directed at forty five degrees relative to the pulling direction, we observe a transition from a seaweed to dendrite morphology as the thermal gradient is lowered, consistent with recent experimental findings. We show that the morphology of crystal structures can be unambiguously characterized through the local interface velocity distribution. We derive semi-empirically a phase diagram for the transition from seaweed to dendrites as a function of thermal gradient and pulling speed. As surface tension anisotropy is increased and aligned with the pulling direction we observe cellular and dendritic arrays directed in the pulling direction. We characterize wavelength selection and obtain a new universal scaling of the wavelength that differs from previous theories.

Type
Research Article
Information
MRS Online Proceedings Library (OPL) , Volume 782: Symposium A – Micro- and Nanosystems , 2003 , A5.65
Copyright
Copyright © Materials Research Society 2004

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Akamatsu, S., Faivre, G., and Ihle, T.. Phys. Rev. E, 51:4751, 1995.Google Scholar
[2] Almgren, R.. SIAM J. Appl. Math., 59:2086, 1999.Google Scholar
[3] Bechhoefer, J. and Libchaber, A.. Phys. Rev. B, 35:1393, 1987.Google Scholar
[4] Ben-Jacob, E., Goldenfeld, N., Kotliar, B.G., and Langer, J.S.. Phys. Rev. Lett., 53:2110, 1984.Google Scholar
[5] Coriell, S. R., McFadden, G. B., and Sekerka, R. F.. Annu. Rev. Mater. Sci., 15:119, 1985.Google Scholar
[6] Flesselles, J.-M., Simon, A.J., and Libchaber, A.J.. Adv. in Phys., 40:1, 1991.Google Scholar
[7] Greenwood, M. and Provatas, N.. Preprint, 2004.Google Scholar
[8] Grossmann, B., Elder, K., Grant, M., and Kosterlitz, M.. Phys. Rev. Lett., 71:3323, 1993.Google Scholar
[9] Hunt, J.D. and Jackson, K.A.. Metall. Trans., 236:843, 1966.Google Scholar
[10] Ihle, T. and Muller-Krumbhaar, H.. Phys. Rev. Lett., 70:3083, 1993.Google Scholar
[11] Karma, A.. Phys. Rev. Lett, 87:115701, 2001.Google Scholar
[12] Karma, A. and Rappel, W.-J.. Phys. Rev. E, 53:3017, 1995.Google Scholar
[13] Kessler, D. A. and Levine, H.. Phys. Rev. A., 31:1712, 1985.Google Scholar
[14] Kessler, D. A. and Levine, H.. Phys. Rev. A., 39:3041, 1989.Google Scholar
[15] Kirkaldy, J. S., Liu, L. X., and Kroupa, A.. Acta Metall. Mater., 43:2905, 1995.Google Scholar
[16] Kopczynski, P., Rappel, W.-J., and Karma, A.. Phys. Rev. E, 55:1282, 1997.Google Scholar
[17] Kurz, W. and Ficher, D. J.. Acta Metallurgica, 29:11, 1981.Google Scholar
[18] Liu, L. X. and Kirkaldy, J. S.. Acta Metall. Mater., 43:2891, 1995.Google Scholar
[19] Losert, W., Mesquita, O. N., Figueiredo, J.M.A., and Cummins, H.Z.. Phys. Rev. Lett., 81:409, 1998.Google Scholar
[20] Losert, W., Shi, B.Q., and Cummins, H.Z.. Proc. Natl. Acad. Sci. USA, 95:431, 1998.Google Scholar
[21] Losert, W., Shi, B.Q., and Cummins, H.Z.. Proc. Natl. Acad. Sci. USA, 95:439, 1998.Google Scholar
[22] Mullins, W. W. and Sekerka, R. F.. J. Appl. Phys., 34:323, 1963.Google Scholar
[23] Provatas, N. and Dantzig, J.. The Enyclopedia of Materials Science and Technology. World Scientife, Oxford, 2001.Google Scholar
[24] Provatas, N., Dantzig, J., and Goldenfeld, N.. Phys. Rev. Lett., 80:3308, 1998.Google Scholar
[25] Provatas, N., Dantzig, J., and Goldenfeld, N.. J. Comp. Phys., 148:265, 1999.Google Scholar
[26] Provatas, N., Wang, Q., Haataja, M., and Grant, M.. Phys. Rev. Lett., 91, 2003.Google Scholar
[27] Saito, Y., Misbah, C., and Muller-Krumbhaar, H.. Phys. Rev. Lett., 63:2377, 1989.Google Scholar
[28] Trivedi, R. and Kurz, W.. Acta metall. mater., 42:15, 1994.Google Scholar
[29] Trivedi, R. and Somboonsuk, K.. Materials Science and Engineering, 65:65, 1984.Google Scholar
[30] Provatas, N. (unpublished).Google Scholar
[31] Utter, B., Ragnarsson, R., and Bodenschatz, E.. Phys. Rev. Lett., 86:4604, 2001.Google Scholar