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The formation of patterns in non-equilibrium growth

Abstract

Crystal growth under non-equilibrium conditions can give rise to complex patterns which are generically similar to those found in processes such as viscous fingering, aggregation and electrochemical deposition. Recent theoretical understanding focuses on the interplay between the macroscopic driving force associated with the phase transition and the microscopic interfacial dynamics.

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References

  1. Kepler, J. De Nive Sexangula Godfrey Tampach (Frankfurt am Main, 1611).

    Google Scholar 

  2. Thompson, D. W. On Growth and Form (Cambridge University Press, 1944).

    Google Scholar 

  3. Fife, P. C. Mathematical Aspects of Reacting and Diffusing Systems (Springer, New York, 1979).

    Book  Google Scholar 

  4. Langer, J. S. Rev. mod Phys. 52, 1–28 (1980).

    Article  CAS  ADS  Google Scholar 

  5. Woodruff, D. P. The Solid-Liquid Interface (Cambridge University Press, 1973).

    Google Scholar 

  6. Ben-Jacob, E., Deutscher, G., Garik, P., Goldenfeld, N. D. & Lereah, Y. Phys. Rev. Lett. 57, 1903–1906 (1986).

    Article  CAS  ADS  Google Scholar 

  7. Pietronero, L. & Toscotti, E. (eds) Fractals in Physics (North Holland, Amsterdam, 1985).

  8. Feder, J. Fractals (Plenum, New York, 1988).

    Book  Google Scholar 

  9. Vicsek, T. (ed.) Fractal Growth Phenomena (World Scientific, New York, 1989).

  10. Nakaya, U. Snow Crystals (Harvard University Press, 1954).

    Book  Google Scholar 

  11. Ivantsov, G. P. Dokl. Akad. Nauk. SSSR 58, 567–569 (1947).

    Google Scholar 

  12. Glicksman, M. E., Shaefer, R. J. & Ayers, J. D. Metall. Trans. A7, 1747–1759 (1976).

    Article  Google Scholar 

  13. Oldfield, W. Mater. Sci. Engng 11, 211–218 (1973).

    Article  CAS  Google Scholar 

  14. Langer, J. S. & Muller-Krümbhaar, H. Acta metall. 26, 1681–1687 (1978); 26, 1689–1695 (1978); 26, 1697–1708 (1978).

    Article  CAS  Google Scholar 

  15. Ben-Jacob, E., Goldenfeld, N. D., Langer, J. S. & Schön, G. Physica D12, 245–252 (1984); Phys. Rev. Lett. 51, 1930–1932 (1983); Phys. Rev. A29, 330–340 (1984).

    Google Scholar 

  16. Hele-Shaw, H. S. Nature 58, 34–36 (1898).

    Article  ADS  Google Scholar 

  17. Paterson, L. J. Fluid Mech. 113, 513–529 (1981); Phys. Rev. Lett. 52, 1621–1625 (1984).

    Article  ADS  Google Scholar 

  18. Maher, J. V. Phys. Rev. Lett. 54, 1498–1501 (1985).

    Article  CAS  ADS  Google Scholar 

  19. Bensimon, D., Kadanoff, L. P., Liang, S., Shraiman, B. I. & Tang, C. Rev. mod. Phys. 58, 977–999 (1986).

    Article  ADS  Google Scholar 

  20. Sander, L. M. Nature 322, 789–793 (1986).

    Article  ADS  Google Scholar 

  21. Mullins, W. W. & Sekerka, R. F. J. appl. Phys. 34, 323–329 (1963); 35, 444–451 (1964).

    Article  CAS  ADS  Google Scholar 

  22. Nittman, J., Daccord, G. & Stanley, H. E. Nature 314, 141–144 (1985).

    Article  ADS  Google Scholar 

  23. Måløy, K. J., Feder, J. & Jossang, J. Phys. Rev. Lett. 55, 2681–2684 (1985).

    Article  Google Scholar 

  24. Witten, T. A. & Sander, L. M. Phys. Rev. Lett. 47, 1400–1403 (1981); Phys. Rev. B27, 5686–5697 (1983).

    Article  CAS  ADS  Google Scholar 

  25. Meakin, P. Phys. Rev. A27, 604–607 (1983); A27, 1495–1507 (1983).

    Article  ADS  Google Scholar 

  26. Buka, A. & Palffy-Muhoray, P. Phys. Rev. A36, 1527–1529 (1987).

    Article  CAS  ADS  Google Scholar 

  27. Schwartz, L. Phys. Fluids 29, 3086–3088 (1986).

    Article  ADS  Google Scholar 

  28. Ben-Jacob, E., Garik, P., Mueller, T. & Grier, D. Phys. Rev. A38, 1370–1380 (1988).

    Article  CAS  ADS  Google Scholar 

  29. Grier, D., Kessler, D. A. & Sander, L. M. Phys. Rev. Lett. 59, 2315–2318 (1987).

    Article  CAS  ADS  Google Scholar 

  30. Sawada, Y., Dougherty, A. & Gollub, J. P. Phys. Rev. Lett. 56, 1260–1263 (1986).

    Article  CAS  ADS  Google Scholar 

  31. Grier, D., Ben-Jacob, E., Clarke, R. & Sander, L. M. Phys. Rev. Lett. 56, 1264–1267 (1986).

    Article  CAS  ADS  Google Scholar 

  32. Tirmizi, S. H. & Gill, W. N. J. Cryst. Growth 96, 277–292 (1989).

    Article  CAS  ADS  Google Scholar 

  33. Goldenfeld, N. D. J. Cryst. Growth 84, 601–608 (1987).

    Article  CAS  ADS  Google Scholar 

  34. May, S. E. & Maher, J. V. Phys. Rev. A40, 1723–1726 (1989).

    Article  CAS  ADS  Google Scholar 

  35. Sarkar, S. & Jasnow, D. Phys. Rev. A39, 5299–5307 (1989).

    Article  CAS  ADS  Google Scholar 

  36. Garik, P. et al. Phys. Rev. Lett. 62, 2703–2706 (1989).

    Article  CAS  ADS  Google Scholar 

  37. Ben-Jacob, E. et al. Phys. Rev. Lett. 55, 1315–1318 (1985).

    Article  CAS  ADS  Google Scholar 

  38. Chan, S. K., Reimer, H. H. & Kahlweit, M. J. Cryst. Growth 32, 303–315 (1976).

    Article  CAS  ADS  Google Scholar 

  39. Kessler, D. A., Koplik, J. & Levine, H. Phys. Rev. A30, 3161–3174 (1984).

    Article  CAS  ADS  Google Scholar 

  40. Ben-Jacob, E., Goldenfeld, N. D., Kotliar, B. G. & Langer, J. S. Phys. Rev. Lett. 53, 2110–2113 (1984).

    Article  CAS  ADS  Google Scholar 

  41. Kessler, D. A., Koplik, J. & Levine, H. Adv. Phys. 37, 255–339.

  42. Langer, J. S. in Chance and Matter Les Houches Summer School (eds Souletie, J., Vannimenus, J. & Stora, R.) (1987).

    Google Scholar 

  43. Langer, J. S. Science 243, 1150–1155 (1989).

    Article  CAS  ADS  Google Scholar 

  44. Saito, Y., Goldbeck-Wood, G. & Muller-Krumbhaar, H. Phys. Rev. Lett. 58, 1541–1544 (1987).

    Article  CAS  ADS  Google Scholar 

  45. Dougherty, A., Kaplan, P. D. & Gollub, J. P. Phys. Rev. Lett. 58, 652–658 (1987); Phys. Rev. A38, 3043–3053 (1988).

    Article  ADS  Google Scholar 

  46. Pieters, R. & Langer, J. S. Phys. Rev. Lett. 56, 1948–1951 (1987).

    Article  ADS  Google Scholar 

  47. Barber, M., Barbieri, A. & Langer, J. S. Phys. Rev. A36, 3340–3349 (1987).

    Article  CAS  ADS  Google Scholar 

  48. Martin, O. & Goldenfeld, N. Phys. Rev. A35, 1382–1390 (1987).

    Article  CAS  ADS  Google Scholar 

  49. Raz, E., Lipson, S. G. & Polturek, E. Phys. Rev. A40, 1088–1095 (1989).

    Article  CAS  ADS  Google Scholar 

  50. Sander, L. M., Ramanlal, P. & Ben-Jacob, E. Phys. Rev. A32, 3160–3163 (1985).

    Article  CAS  ADS  Google Scholar 

  51. Hecker, N. thesis, Univ. of Michigan (1988).

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Ben-Jacob, E., Garik, P. The formation of patterns in non-equilibrium growth. Nature 343, 523–530 (1990). https://doi.org/10.1038/343523a0

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