Math @ Duke

Publications [#304585] of JianGuo Liu
Papers Published
 Li, B; Liu, JG, Epitaxial growth without slope selection: Energetics, coarsening, and dynamic scaling,
Journal of Nonlinear Science, vol. 14 no. 5
(October, 2004),
pp. 429451, Springer Nature, ISSN 09388974 [doi]
(last updated on 2019/06/26)
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 Δ h 2, which represents the surface diffusion, andlog (1 + ∇ 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 ε 2 of Δ h 2 with ε = 1/L and L the linear size of the underlying system, and study the energy asymptotics in the large system limit ε → 0. We show that global minimizers of the freeenergy functional exist for each ε > 0, the L 2norm of the gradient of any global minimizer scales as O(1/ε), and the global minimum energy scales as O( log ε). 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. © 2005 Springer.


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