Math @ Duke

Publications [#246959] of JianGuo Liu
Papers Published
 Li, B; Liu, JG, Eptaxial growth without slope selection: energetics, coarsening, and dynamic scaling,
J. Nonlinear Sci., vol. 14 no. 5
(2004),
pp. 429451, ISSN 09388974 [doi]
(last updated on 2019/04/23)
Abstract: We study a continuum model for epitaxial growth of thin films in which the slope of mound structure of film surface increases. This model is a diffusion equation for the surface height profile h which is assumed to satisfy the periodic boundary condition. The equation happens to possess a Liapunov or "freeenergy" functional. This functional consists of the term Î” h 2, which represents the surface diffusion, andlog (1 + âˆ‡ h 2), which describes the effect of kinetic asymmetry in the adatom attachmentdetachment. We first prove for large time t that the interface widththe standard deviation of the height profileis bounded above by O(t 1/2), the averaged gradient is bounded above by O(t 1/4), and the averaged energy is bounded below by O(log t). We then consider a small coefficient Îµ 2 of Î” h 2 with Îµ = 1/L and L the linear size of the underlying system, and study the energy asymptotics in the large system limit Îµ â†’ 0. We show that global minimizers of the freeenergy functional exist for each Îµ > 0, the L 2norm of the gradient of any global minimizer scales as O(1/Îµ), and the global minimum energy scales as O( log Îµ). The existence of global energy minimizers and a scaling argument are used to construct a sequence of equilibrium solutions with different wavelengths. Finally, we apply our minimum energy estimates to derive bounds in terms of the linear system size L for the saturation interface width and the corresponding saturation time. Â© 2005 Springer.


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