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Rayleigh Instability. Inkjet Printing. From Pimbley et al. [1977]. Separation of a fluid plane into drops.. Cell Growth amid Directional Solidification. From Kurowski et al. [1989]. Separation of fluid sections into drops amid cementing of CBr4.. Insecurity of Rod Morphology During Monotectic Growth.

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

Rayleigh Instability

Inkjet Printing From Pimbley et al. [1977]. Separation of a fluid stream into drops.

Cellular Growth amid Directional Solidification From Kurowski et al. [1989]. Separation of fluid depressions into drops amid cementing of CBr 4 .

Instability of Rod Morphology During Monotectic Growth From Majumdar et al. [1996]. Separation of adjusted bars into drops amid helpful monotectic development of Zinc-Bismuth. .

Nanobridge From Kondo et al. [1997]. Unsupported extension framed by utilizing electron bar illumination as a part of a ultrahigh vacuum electron magnifying instrument.

Quantum Wires From Chen et al. [2000]. STM topographs indicating ErSi 2 (011) nanowires developed on a level Si(001) substrate. The Si patios increment in range from dark blue to green.

Possible Reasons for Enhanced Stability Quantum impacts (Kassubek et al. [2001]). Versatile impacts with substrate (Chen et al. [2000]) Stabilization by contact point (McCullum et al. [1996]) Radial warm slopes (McFadden et al. [1993]) Surface vitality 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 3-D Equilibrium Shapes for - 1/18 < 4 <1/12 High-Symmetry Orientations: [001], [011], [111]

Cubic Material

Asymptotics for | 4 |<< 1

Numerics SLEIGN2: Associated Sturm–Liouville Solver Spectral Decomposition with RS (a genuine symmetric eigenvalue schedule)

[001] Orientation 4 = 1/12 0 - 1/18 < 4 < 1/12 1 2

[011] Orientation - 1/18 < 4 < 1/12

011 Orientation 0 1 2

111 Orientation

Generalized Gauss Curvature

Conclusions Anisotropic surface vitality assumes a critical part in the soundness of a pole. Both the extent and indication of the anisotropy figure out if the commitment advances or smothers the Rayleigh unsteadiness. Distinctive cubic introductions respond contrastingly to the surface strain anisotropy. Future Work Missing introductions Contact edges Elastic impacts

P.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-gathered development of epitaxial erbium disilicide nanowires of silicon(001), App. Phys. Lett., Vol. 76, No. 26 (2000), 4004- - 4006. M.G. Backwoods and Q. Wang, Anisotropic microstructure-incited diminishment of the Rayleigh insecurity for fluid crystalline polymers, Phys. Lett. A, 245 (1998) 518- - 526. J.W. Cahn, Stability of bars with anisotropic surface free vitality, Scripta Metall. 13 (1979) 1069-1071. F. Kassubek, C.A. Stafford, H. Grabert, and R.E. Goldstein, Quantum concealment of the Rayleigh precariousness in nanowires, Nonlinearity 14 (2001) 167- - 177. P. Kurowski, S. de Cheveigne, G. Faivre, and C. Guthmann, Cusp insecurity in cell development, J. Phys. (Paris) 50 (1989) 3007-3019. Y. Kondo and K. Takayanagi, Gold nanobridge balanced out 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. Tangle. Trans. A, Vol 27A, July (1996) 2053- - 2057. M.S. McCallum, P.W. Voorhees, M.J. Miksis, S.H. Davis, and H. Wong, Capillary insecurities in strong thin movies: Lines, J. Appl. Phys. 79 (1996) 7604-7611. G.B. McFadden, S.R. Coriell, and R.F. Sekerka, Effect of surface strain anisotropy on cell morphologies, J. Precious stone Growth 91 (1988) 180- - 198. G.B. McFadden, S.R. Coriell, and B.T. Murray, The Rayleigh precariousness for a round and hollow gem soften interface, in Variational and Free Boundary Problems , (ed. A. Friedman and J. Spruck), Vol. 53 (1993) pp. 159-169. References

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