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Asymptotic solutions for the equilibrium crystal shape with small corner energy regularization
1Department of Mathematics, University at Buffalo, Buffalo, New York 14260-2900, USA.
Summary
This study analyzes how corner energy regularization affects solid particle shapes. The findings validate using regularization in calculations and provide an exact solution for wedge shapes.
Area of Science:
- Materials Science
- Solid State Physics
- Computational Modeling
Background:
- Dynamic models of facet formation commonly use surface energy regularization with a corner energy term.
- Understanding the impact of this regularization on equilibrium shapes is crucial for accurate modeling.
Purpose of the Study:
- To investigate the effect of corner energy regularization on the equilibrium shape of 2D solid particles.
- To derive an explicit solution for corner shapes under regularization.
- To validate the use of regularization in equilibrium calculations.
Main Methods:
- Application of matched asymptotic expansions to determine corner shapes.
- Analysis of surface energy anisotropy models.
Main Results:
- An explicit solution for the corner shape with regularization was determined.
- For certain anisotropy models, regularized solutions converge to sharp-corner results as regularization diminishes.
- An exact solution for the equilibrium shape of a regularized semi-infinite wedge was obtained.
Conclusions:
- The study validates the use of corner energy regularization in numerical simulations for equilibrium shape problems.
- The derived solutions provide theoretical support for computational approaches in materials science.
- The findings contribute to a deeper understanding of surface phenomena and particle morphology.