Journal
PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES
Volume 465, Issue 2105, Pages 1337-1359Publisher
ROYAL SOC
DOI: 10.1098/rspa.2008.0385
Keywords
thin film; quantum dots; interfacial anisotropy; phase field; Cahn-Hilliard equations; Willmore regularization
Categories
Funding
- Division Of Materials Research
- Direct For Mathematical & Physical Scien [0835653] Funding Source: National Science Foundation
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We present a new phase-field model for strongly anisotropic crystal and epitaxial growth using regularized, anisotropic Cahn Hilliard-type equations. Such problems arise during the growth and coarsening of thin films. When the anisotropic surface energy is sufficiently strong, sharp corners form and unregularized anisotropic Cahn Hilliard equations become ill-posed. Our models contain a high-order Willmore regularization, where the square of the mean curvature is added to the energy, to remove the ill-posedness. The regularized equations are sixth order in space. Akey feature of our approach is the development of a new formulation in which the interface thickness is independent of crystallographic orientation. Using the method of matched asymptotic expansions, we show the convergence of our phase-field model to the general sharp-interface model. We present two- and three-dimensional numerical results using an adaptive, nonlinear multigrid finite-difference method. We find excellent agreement between the dynamics of the new phase-field model and the sharp-interface model. The computed equilibrium shapes using the new model also match a recently developed analytical sharp- interface theory that describes the rounding of the sharp corners by the Willmore regularization.
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