Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Letter
  • Published:

Interface pattern formation in nonlinear dissipative systems

Nature Materials volume 1, pages 157–159 (2002)Cite this article

Abstract

The problem of interface pattern selection in nonlinear dissipative systems is critical in many fields of science, occurring in physical, chemical and biological systems. One of the simplest pattern formations is the Saffman–Taylor finger pattern1 that forms when a viscous fluid is displaced by a less viscous fluid. Such finger-shaped patterns have been observed in distinctly different fields of science2,3,4 (hydrodynamics, combustion and crystal growth) and this has led to a search for a unified concept of pattern formation, as first proposed by the classic work of D'arcy Thomson5. Two-dimensional finger-shaped patterns, observed in flame fronts6 and the ensembled average shape of the diffusion-limited aggregation pattern, have been shown to be similar to Saffman–Taylor finger shapes7,8. Here we present experimental studies that establish that the cell shapes formed during directional solidification of alloys can be described by the form of the Saffman–Taylor finger shape equation when a second phase is present in the intercellular region.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Figure 1: A comparison of the Saffman–Taylor finger shape with the shape of a cell in directional solidification.
Figure 2: Shapes of cells in an array in directional solidification.

Similar content being viewed by others

References

  1. Saffman, P.G. & Taylor, G.I. The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. Lond. A 245, 312–329 (1958).

    Article  CAS  Google Scholar 

  2. Pelce, P. Dynamics of Curved Fronts (Academic, Oxford, 1988)

  3. Kessler, D., Koplik, J. & Levine, H. Pattern selection in fingered growth phenomena. Adv. Phys. 37(3), 255–339 (1988).

    Article  Google Scholar 

  4. Langer, J.S. in Chance and Matter (ed. Souleite, J.) (North Holland, Amsterdam, 1987).

    Google Scholar 

  5. Thomson, D.W. On Growth and Form (Cambridge Univ. Press, 1942).

  6. Britten, J.A., Krantz W.B., Visser, W. & Gunn, R.D. Theoretical and Experimental Studies of Reverse Combustion (Internal report, Univ.of Colorado, Boulder, 1985).

    Google Scholar 

  7. Couder, Y, Argoul, F, Arnéodi, A., Maurer, J. & Ribaud, M. Statistical properties of the fractal dendrites and anisotropic diffusion-limited aggregates. Phys. Rev. A 42, 3499–3503 (1990).

    Article  CAS  Google Scholar 

  8. Arneodo, A., Argoul, F., Couder, Y. & Ribaud, M. Anisotropic Laplacian growths: from diffusion-limited aggregates to dendrite fractals. Phys. Rev. Lett. 66, 2332–2335 (1991).

    Article  CAS  Google Scholar 

  9. Somboonsuk, K., Mason, J.T. & Trivedi, R. Interdendritic spacing: Part 1. Experimental studies. Metall. Trans. 15A, 967–975 (1984).

    Article  CAS  Google Scholar 

  10. Pelcé, P. & Pumir, A. Cell shape in directional solidification in the small Peclet number limit. J. Cryst. Growth 73, 337–341 (1985).

    Article  Google Scholar 

  11. Billia, B., Jamgotchian, H. & Capella, L. Pattern formation during directional solidification. J. Cryst. Growth, 82, 747–756 (1987).

    Article  CAS  Google Scholar 

  12. Billia, B. & Trivedi, R. Pattern formation in crystal growth. In Handbook of Crystal Growth (2b), 993 (North Holland, Amsterdam, 1995).

    Google Scholar 

  13. Weeks, J.D., Van Sarloos, W. & Grant, M. Stability and shapes of cellular profiles in directional solidification: expansion and matching methods. J. Cryst. Growth 112, 244–282 (1991).

    Article  CAS  Google Scholar 

  14. Karma, A. & Pelce, P. Stability of an array of deep cells in directional solidification. Phys Rev A, 41, 6741–6748 (1990).

    Article  CAS  Google Scholar 

  15. Dombre, T. & Hakim, V. Saffman–Taylor fingers and directional solidification at low velocity. Phys. Rev. A 36(6), 2811–2817 (1987).

    Article  Google Scholar 

  16. Trivedi, R., Shen, Y. & Liu, S. A unique correlation among cellular microstructural length scales in directionally solidified binary systems. In Proc. Int. Conf. Advances in Materials and Materials Processing (ed. Chakraborty, N. & Chatterjee, U.) 42–49 (Tata McGraw-Hill, 2001).

    Google Scholar 

  17. Levine, H. & Tu, Y. Mean-field diffusion-limited aggregation and the Saffman–Taylor problem in three dimensions. Phys. Rev. A 45(2), 1044–1052 (1992).

    Article  Google Scholar 

  18. Brener, E. Non-axisymmetric patterns in the Saffman–Taylor problem and in the three dimension directional solidification at low velocity. Phys. Rev. E 48, 4437–4443 (1993).

    Article  CAS  Google Scholar 

  19. Scheil, E. Bemerkungen zur Schichtkristasllbildung. Z. Metallkunde 34, 70–72 (1942).

    Google Scholar 

Download references

Acknowledgements

This work was supported by the Office of Microgravity, NASA, and funded through the Marshall Space Flight Center.

Author information

Authors and Affiliations

  1. Ames Laboratory (US-DOE) and Department of Materials Science and Engineering, Metals and Ceramics Sciences, Iowa State University, Ames, 50011, Iowa, USA

    Rohit Trivedi, Shan Liu & Scott Williams

Authors
  1. Rohit Trivedi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shan Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Scott Williams
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rohit Trivedi.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Trivedi, R., Liu, S. & Williams, S. Interface pattern formation in nonlinear dissipative systems. Nature Mater 1, 157–159 (2002). https://doi.org/10.1038/nmat749

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nmat749

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing