Willmore regularized sharp-interface model for strongly anisotropic solid-state dewetting with axisymmetric geometry: modeling and simulation

TL;DR

The paper addresses axisymmetric solid-state dewetting under strong crystalline anisotropy, where the evolution can be ill-posed. It introduces a Willmore-regularized energy $W(\mathcal S)$ and derives a sharp-interface model for the generating curve, including a dimensionless evolution equation $r\,\vec X_t\cdot\vec n = (r\,\mu_s)_s$ with a detailed chemical potential $\mu$ that accounts for anisotropy and Willmore regularization. A two-pronged geometric reformulation via $\boldsymbol{B}_q(\theta)$ yields an equivalent system, which is then cast into a variational form and discretized by a structure-preserving parametric finite element method that ensures both volume conservation and energy stability. Extensive numerical experiments demonstrate second-order convergence, energy dissipation, volume preservation, and improved mesh quality, validating the method’s accuracy and stability for long-time SSD simulations with strong anisotropy. The approach provides a rigorous, robust framework for simulating axisymmetric SSD and offers a foundation for studying basins of attraction and pinch-off phenomena in highly anisotropic materials.

Abstract

In this work, we consider the three-dimensional solid-state dewetting with strongly anisotropic surface energy, assuming an axisymmetric morphology of the thin film. However, when surface energy exhibits strong anisotropy, certain orientations may be missing from the equilibrium shapes, which will lead to an ill-posed governing equation. By incorporating the Willmore energy, we define a regularized total free energy and rigorously derive a sharp-interface model based on thermodynamic variations. We further develop a numerical scheme for the sharp-interface model that can preserve two important structural properties, including both the volume-conservation and energy-stability laws. We conclude by presenting a series of numerical simulations that illustrate the accuracy and structure-preserving properties. More importantly, extensive numerical simulations clearly demonstrate that our schemes can significantly enhance mesh quality, which is beneficial for long-term computations.

Figures (12)

• Figure 1: A schematic illustration of the solid-state dewetting (left panel) a toroidal thin film on a flat substrate ; The cross-section of an axis-symmetric thin film in the cylindrical coordinate system $(r, z)$, $r_i$ and $r_o$ represent the radius of inner contact line and outer contact line respectively.
• Figure 2: Plot of the numberical errors at $\Delta t_m = 1$(left panel) and $\Delta t_m$ = 2(right panel) for 4-fold anisotropy. The initial data $\vec{X}^0(\rho) = (10 + \cos(\pi\rho), \sin(\pi\rho))$, and the parameters are selected as $\eta = 100$, $\sigma = -0.6$, $\varepsilon = 0.01$.
• Figure 3: Plot of numberical errors at $\Delta t_m = 1$(left panel) and $\Delta t_m$ = 2(right panel) for 4-fold anisotropy. The initial data $\vec{X}^0(\rho) = (10 + \cos(\pi\rho), \sin(\pi\rho))$and the parameters are selected as $\eta = 100$, $\sigma = -0.6$, $\beta = 0.07$.
• Figure 4: The time history of the energy ratio $E(t)/E(0)$ employing structure-preserving method with $\beta = 0.07$ (left panel) and $\beta = 0.1$ (right panel). The initial data $\vec{X}^0(\rho) = (10 + \cos(\pi\rho), \sin(\pi\rho))$, and the parameters are selected as $\eta = 100$, $\sigma = -0.6$, $\varepsilon = 0.01$.
• Figure 5: The time history of the energy ratio $E(t)/E(0)$ employing structure-preserving method with $\beta = 0.07$ (left panel) and $\beta = 0.1$ (right panel). We choose the initial data $\vec{X}^0(\rho) = (\cos(\pi\rho/2), \sin(\pi\rho/2))$. The parameters are selected as $\eta = 100$, $\sigma = -0.6$, $\varepsilon = 0.005$.

Theorems & Definitions (11)

• Remark 1
• Theorem 4.1
• proof
• Lemma 5.1
• proof
• Remark 2
• Theorem 5.2
• proof
• Theorem 5.3
• proof