Abstract
The SMB equation describing nanoscale spontaneous patterning is studied both analytically and numerically. In contradiction to the claim in the original SMB paper [D. Srolovitz, A. Mazor, B. Bukiet, J. Vac. Sci. Technol. A6(4) (1988), 2371--2380.] that some steady states are stable, we found that all the steady states are unstable. A dynamical system reason for this is given. We also found that typical small initial data solutions undergo an exponential growth followed by an almost linear growth. Such a feature is consistent with the experimental data in the paper [J. Erlebacher et al., J. Vac. Sci. Technol., A18(1) (2000), 115--120, Figure 3]. On the other hand, we never observed the decay portion of the numerical solution reported in this paper. We invent an elegant energy principle which supports our findings.
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References
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D. Srolovitz, A. Mazor and B. Bukiet, Analytical and numerical modeling of columnar evolution in thin films. J. Vac. Sci. Technol. A6(4):2371 (1988).
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Feng, Z.C., Li, Y.C. Nanoscale Spontaneous Patterning. J Stat Phys 123, 741–751 (2006). https://doi.org/10.1007/s10955-006-9083-1
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DOI: https://doi.org/10.1007/s10955-006-9083-1