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

Mathematical and Computational Sciences DivisionNational Institute of Standards and Technology

K.F. Gurski and G.B. McFaddenK.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)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

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 < 1/120

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

ReferencesP.B. Bailey, W.N. Everitt, and A. Zettl, Algorithm 810: The SLEIGN2 Sturm-Liouville code, ACM T Math Software 27: (2) Jun 2001 143--192.

Y. Chen, D.A.A. Ohlberg, G. Medeiros-Ribeiro, Y.A. Chang, and R.S. Williams, Self-assembled growth of epitaxial erbium disilicide nanowires of silicon(001), App. Phys. Lett., Vol. 76, No. 26 (2000), 4004--4006.

M.G. Forest and Q. Wang, Anisotropic microstructure-induced reduction of the Rayleigh instability for liquid crystalline polymers, Phys. Lett. A, 245 (1998) 518--526.

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.

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) 2053--2057.

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) 7604-7611.

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.

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. 159-169.