A regularized phase-field model for faceting in a kinetically controlled crystal growth

Area of Science:

• Materials Science
• Crystallography
• Computational Physics

Background:

• Strongly anisotropic crystals develop corners at equilibrium due to negative surface stiffness.
• Traditional sharp-interface models use curvature-dependent terms to regularize corner dynamics.
• Diffuse interface models offer an alternative for regularizing these phenomena.

Purpose of the Study:

• To develop and validate a diffuse interface phase-field model for strongly anisotropic crystals.
• To regularize the model using an approximation of the Willmore energy.
• To investigate equilibrium corner properties and growth dynamics.

Main Methods:

• Phase-field modeling using the Allen-Cahn equation for kinetically controlled growth.
• Method of matched asymptotic expansions to connect with sharp-interface theory.
• Analysis of the stress tensor to derive forces on the diffuse interface.

Main Results:

• The model converges to Herring's sharp-interface theory.
• Equilibrium corner properties were examined using the stress tensor.
• Phase-field simulations confirmed a parabolic coarsening regime (mean facet length ~ t^1/2).
• A specific coarsening mechanism involving merging valleys was observed.

Conclusions:

• The diffuse interface model provides a robust framework for studying anisotropic crystal growth.
• The model successfully reproduces key features of sharp-interface theories, including corner behavior and coarsening dynamics.
• This approach offers insights into the fundamental mechanisms governing crystal morphology evolution.