A sharp-interface approach for simulating solid-state dewetting of thin films with double-bubble structure

TL;DR

This work addresses SSD of double-bubble thin films with anisotropic interfacial energies by formulating a sharp-interface, energy-variational model founded on the Cahn-Hoffman vector $\boldsymbol{\xi}$. The authors derive the evolution equations $\partial_t\boldsymbol{X}_j=\partial_{s_j s_j}\mu_j\boldsymbol{n}_j$ with $\mu_j=-\partial_{s_j}\boldsymbol{\xi}_j^{\perp}\cdot\boldsymbol{n}_j$ and provide a symmetric, conservative variational framework using $\boldsymbol{Z}_{K,j}(\boldsymbol{n}_j)$, ensuring area conservation and energy dissipation. A structure-preserving PFEM (SP-PFEM) is proposed, with a discrete scheme that preserves area ($A^m=A^0$) and monotonically decreases energy under suitable conditions; an ES-PFEM variant is discussed that is linear but sacrifices area conservation. Numerical results confirm convergence, mesh-quality preservation, and the ability to capture equilibrium shapes and pinch-off events across isotropic, weakly anisotropic, and strongly anisotropic regimes, highlighting the framework’s robustness for multi-film SSD problems.

Abstract

We develop a sharp-interface model for solid-state dewetting of double-bubble thin films using an energy variational approach based on a newly proposed interfacial energy. This model characterizes the dynamic evolution of interfaces in double-bubble thin films, a process primarily governed by surface diffusion and junction/contact points migration, and fundamentally distinct from the behavior observed in a single thin film. Subsequently, a structure-preserving parametric finite element approximation is developed for the sharp-interface model, which can preserve both area conservation and energy stability. Extensive numerical experiments are presented to demonstrate the convergence, structure-preserving properties, and superior mesh quality of the proposed method. Additionally, we investigate several specific evolution processes, including the equilibrium shapes of double-bubble thin films and the pinch-off dynamics of long islands.

Figures (18)

• Figure 1: A schematic description of the SSD.
• Figure 2: Plot of numberical errors employing SP-PFEM at $\sigma_1 =\sigma_2 = -0.7$ (left panel) and $\sigma_1 =\sigma_2 = 0.7$ (right panel) for $2$-fold anisotropy. The parameters are selected as $\eta_1 =\eta_2 =\eta_3 = 100$, and $t_m = 1$.
• Figure 3: Plot of numberical errors employing SP-PFEM at $\sigma_1 =\sigma_2 = -0.7$ (left panel) and $\sigma_1 =\sigma_2 = 0.7$ (right panel) for $2$-fold anisotropy. The parameters are selected as $\eta_1 =\eta_2 =\eta_3 = 100$, and $t_m = 2$.
• Figure 4: Plot of numberical errors employing SP-PFEM at $\sigma_1 =\sigma_2 = -0.7$ (left panel) and $\sigma_1 =\sigma_2 = 0.7$ (right panel) for $4$-fold anisotropy. The parameters are selected as $\eta_1 =\eta_2 =\eta_3 = 100$, and $t_m = 1$.
• Figure 5: Plot of numberical errors employing SP-PFEM at $\sigma_1 =\sigma_2 = -0.7$ (left panel) and $\sigma_1 =\sigma_2 = 0.7$ (right panel) for $4$-fold anisotropy. The parameters are selected as $\eta_1 =\eta_2 =\eta_3 = 100$, and $t_m = 2$.

Theorems & Definitions (17)

• Theorem 2.1
• proof
• Remark 1
• Proposition 2.1
• proof
• Proposition 3.1
• proof
• Remark 2
• Proposition 4.1
• proof