Steady needle growth with 3-D anisotropic surface tension

Abstract

The effect of the anisotropic interfacial energy on dendritic growth has been an important subject, and has preoccupied many researchers in the field of materials science and condensed matter physics. The present paper is dedicated to the study of the effect of full 3-D anisotropic surface tension on the steady state solution of dendritic growth. We obtain the analytical form of the first order approximation solution in the regular asymptotic expansion around the Ivantsov’s needle growth solution, which extends the steady needle growth solution of the system with isotropic surface tension obtained by Xu and Yu (J. J. Xu and D. S. Yu, J. Cryst. Growth, 1998, 187: 314; J. J. Xu, Interfacial Wave Theory of Pattern Formation: Selection of Dendrite Growth and Viscous Fingering in a Hele-Shaw Flow, Berlin: Springer-Verlag, 1997).The solution is expanded in the general Laguerre series in any finite region around the needle-tip, and it is also expanded in a power series in the far field behind the tip. Both solutions are then numerically matched in the intermediate region. Based on this global valid solution, the dependence of Peclet number Pe and the interface’s morphology on the anisotropy parameter of surface tension as well as other physical parameters involved are determined. On the basis of this global valid solution, we explore the effect of the anisotropy parameter on the Peclet number of growth, as well as the morphology of the interface.

References

1. D. A.Kessler and H.Levine, Phys. Rev. A, 1987, 36(2): 4123. doi: 10.1103/PhysRevA.36.4123
2. D. A.Kessler and H.Levine, Phys. Rev. Lett., 1986, 57: 3069. doi: 10.1103/PhysRevLett.57.3069
3. F.Liu and N.Goldefeld, Phys. Rev. A, 1990, 42(8): 5052. doi: 10.1103/PhysRevA.42.5052
4. E. A.Brener, J. Cryst. Growth, 1990, 99: 165. doi: 10.1016/0022-0248(90)90505-F
5. J. J.Xu, Canad. J. Phys., 1991, 69(7): 789
6. J. J.Xu, Phys. Rev. E, 1996, 53(5): 5031. doi: 10.1103/PhysRevE.53.5051
7. J. J.Xu, Interfacial Wave Theory of Pattern Formation: Selection of DendriteGrowth and Viscous Fingering in a Hele-Shaw Flow, Berlin: Springer-Verlag, 1997
8. M. E.Glicksman and S. P.Marsh, “The Dendrite”in: Handbook of Crystal Growth, Volume 1: Fundamentals, Part B: Transport and Stability,edited by D. T. J.Hurle, Amsterdam, North-Holland: ElsevierScience, 1993
9. K. F.Gurski and Jeff. B.McFadden, Proc. R. Soc. Lond.A, 2003, 459: 1
10. C.Herring, in: Structure and Properties of Solid Surface, editedby R. Gomer and C. S.Smith, University of Chicago, 1952
11. Jeff. B.McFadden, Private communication, Nov., 2007
12. E. A.Brener and V. I.Melnikov, Adv. Phys., 1991, 40: 53. doi: 10.1080/00018739100101472
13. M. Abramovitz and I. A. Stegun (Eds.), Handbook ofMathematical Functions, National Bureau of Standards, 1964
14. J. J.Xu and D. S.Yu, J. Cryst. Growth, 1998, 187: 314. doi: 10.1016/S0022-0248(97)00864-6
15. J. J.Xu, J. Cryst. Growth, 2002, 245: 134. doi: 10.1016/S0022-0248(02)01642-1