# Presentation: 884483

## The Effect of Surface Tension Anisotropy on the Rayleigh Instability in Materials Systems

- Mathematical and Computational Sciences Division
- National Institute of Standards and Technology

- K.F. Gurski and G.B. McFadden

- NASA Microgravity, NSF NIRT (NWU)

- 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

## Rayleigh Instability

## Inkjet Printing

- From Pimbley et al. [1977]. Breakup of a liquid jet into drops.

## Cellular Growth during Directional Solidification

- From Kurowski et al. [1989].  Breakup of liquid  grooves into drops during solidification of CBr4.

## 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..

## Nanobridge

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

## 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.

## 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)

## Anisotropic Gibbs-Thomson Equation

## Cahn-Hoffman Xi-Vector (2-D)

## Cahn-Hoffman Xi-Vector (3-D)

## 2-D Rod from 3-D Equilibrium Shape

## Shape Perturbation

## Surface Energy

## Eigenvalue Problem

- Eigenvalue Problem

## Eigenvalue Problem

## Isotropic Surface Energy

## Ellipsoidal Surface Energy

## Cubic Material

- High-Symmetry Orientations: [001], [011], [111]

- 3-D Equilibrium Shapes for
- -1/18 < 4 <1/12

## Cubic Material

-                    Asymptotics for |4|<< 1

## Numerics

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

## [001] Orientation

- 4 = 1/12

- -1/18 < 4 < 1/12

- 0

- 1

- 2

## [011] Orientation

- -1/18 < 4 < 1/12

## 011 Orientation

- 0

- 1

- 2

## 111 Orientation

## Generalized Gauss Curvature

## Conclusions

- 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

## References

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- J.W. Cahn, Stability of rods with anisotropic surface free energy, Scripta Metall. 13 (1979) 1069-1071.
- F. Kassubek, C.A. Stafford, H. Grabert, and R.E. Goldstein, Quantum suppression of the Rayleigh instability in nanowires, Nonlinearity 14 (2001) 167--177.
- P. Kurowski, S. de Cheveigne, G. Faivre, and C. Guthmann, Cusp instability in cellular growth, J. Phys. (Paris) 50 (1989) 3007-3019.

- Y. Kondo and K. Takayanagi, Gold nanobridge stabilized by surface structure, Phys. Rev. Lett. 79 (1997) 3455-3458.
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- G.B. McFadden, S.R. Coriell, and R.F. Sekerka, Effect of surface tension anisotropy on cellular morphologies, J. Crystal Growth 91 (1988) 180--198.
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