1. The Effect of Surface Tension Anisotropy on the Rayleigh Instability in Materials Systems
K.F. Gurski and G.B. McFadden
Mathematical and Computational Sciences Division
National Institute of Standards and Technology
Introduction to the Rayleigh instability
Anisotropic surface energy
2-D equilibrium shapes
Rayleigh instability for anisotropic surface energy
Conclusions and future work
Thanks to S.H. Davis
NASA Microgravity, NSF NIRT (NWU)

2. Rayleigh Instability

3. Inkjet Printing
From Pimbley et al. [1977]. Breakup of a liquid jet into drops.

4. Cellular Growth during Directional Solidification
From Kurowski et al. [1989]. Breakup of liquid grooves into drops during solidification of CBr4.

5. Instability of Rod Morphology During Monotectic Growth
From Majumdar et al. [1996]. Breakup of aligned rods into drops during cooperative monotectic growth of Zinc-Bismuth..

6. Nanobridge
From Kondo et al. [1997]. Free-standing bridge formed by using electron beam irradiation in an ultrahigh vacuum electron microscope.

7. Quantum Wires
From Chen et al. [2000]. STM topographs showing ErSi2 (011) nanowires grown on a flat Si(001) substrate. The Si terraces increase in height from deep blue to green.

8. Possible Reasons for Enhanced Stability
Quantum effects (Kassubek et al. [2001]).
Elastic effects with substrate (Chen et al. [2000])
Stabilization by contact angle (McCullum et al. [1996])
Radial thermal gradients (McFadden et al. [1993])
Surface energy anisotropy (this work)

9. Anisotropic Gibbs-Thomson Equation

10. Cahn-Hoffman Xi-Vector (2-D)

11. Cahn-Hoffman Xi-Vector (3-D)

12. 2-D Rod from 3-D Equilibrium Shape

13. Shape Perturbation

14. Surface Energy

15. Eigenvalue Problem

16. Eigenvalue Problem

17. Isotropic Surface Energy

18. Ellipsoidal Surface Energy

19. 3-D Equilibrium Shapes for -1/18 < 4 <1/12
Cubic Material
3-D Equilibrium Shapes for
-1/18 < 4 <1/12
High-Symmetry Orientations: [001], [011], [111]

20. Cubic Material

21. Asymptotics for |4|<< 1

22. Numerics SLEIGN2: Associated Sturm–Liouville Solver
Spectral Decomposition with RS (a real symmetric eigenvalue routine)

23. [001] Orientation
4 = 1/12 0
-1/18 < 4 < 1/12 1 2

24. [011] Orientation
-1/18 < 4 < 1/12

25. 011 Orientation 0 1 2

26. 111 Orientation

27. Generalized Gauss Curvature

28. Conclusions Future Work
Anisotropic surface energy plays a significant role in the stability of a rod.
Both the magnitude and sign of the anisotropy determine whether the contribution promotes or suppresses the Rayleigh instability.
Different cubic orientations react quite differently to the surface tension anisotropy.
Future Work
Missing orientations
Contact angles
Elastic effects

29. References
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B. Majumdar and K. Chattopadhyay, The Rayleigh Instability and the Origin of Rows of Droplets in the Monotectic Microstructure of Zinc-Bismuth Alloys, Met. Mat. Trans. A, Vol 27A, July (1996)
M.S. McCallum, P.W. Voorhees, M.J. Miksis, S.H. Davis, and H. Wong, Capillary instabilities in solid thin films: Lines, J. Appl. Phys. 79 (1996)
G.B. McFadden, S.R. Coriell, and R.F. Sekerka, Effect of surface tension anisotropy on cellular morphologies, J. Crystal Growth 91 (1988)
G.B. McFadden, S.R. Coriell, and B.T. Murray, The Rayleigh instability for a cylindrical crystal-melt interface, in Variational and Free Boundary Problems, (ed. A. Friedman and J. Spruck), Vol. 53 (1993) pp
P.B. Bailey, W.N. Everitt, and A. Zettl, Algorithm 810: The SLEIGN2 Sturm-Liouville code, ACM T Math Software 27: (2) Jun
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M.G. Forest and Q. Wang, Anisotropic microstructure-induced reduction of the Rayleigh instability for liquid crystalline polymers, Phys. Lett. A, 245 (1998)
J.W. Cahn, Stability of rods with anisotropic surface free energy, Scripta Metall. 13 (1979)
F. Kassubek, C.A. Stafford, H. Grabert, and R.E. Goldstein, Quantum suppression of the Rayleigh instability in nanowires, Nonlinearity 14 (2001)
P. Kurowski, S. de Cheveigne, G. Faivre, and C. Guthmann, Cusp instability in cellular growth, J. Phys. (Paris) 50 (1989)