Structure-preserving parametric finite element methods for simulating axisymmetric solid-state dewetting problems with anisotropic surface energies

TL;DR

This work tackles axisymmetric solid-state dewetting with anisotropic surface energy by developing structure-preserving PFEMs that conserve volume and ensure energy stability. It introduces two surface-energy matrices derived from anisotropy functions and two corresponding weak formulations, enabling two classes of PFEMs with proven volumetric and energetic properties. Further, it presents alternative weak formulations to improve mesh quality and linear/volume-preserving variants, all validated via extensive numerical tests across multiple anisotropy types, including BGN. The methods provide accurate, stable, and efficient simulations of SSD in axisymmetric settings with broad applicability to anisotropic energies, offering a solid foundation for reliable micro-/nano-scale patterning simulations.

Abstract

Solid-state dewetting (SSD), a widespread phenomenon in solid-solid-vapor system, could be used to describe the accumulation of solid thin films on the substrate. In this work, we consider the sharp interface model for axisymmetric SSD with anisotropic surface energy. By introducing two types of surface energy matrices from the anisotropy functions,we aim to design two structure-preserving algorithms for the axisymmetric SSD. The newly designed schemes are applicable to a broader range of anisotropy functions, and we can theoretically prove their volume conservation and energy stability. In addition, based on a novel weak formulation for the axisymmetric SSD, we further build another two numerical schemes that have good mesh properties. Finally, numerous numerical tests are reported to showcase the accuracy and efficiency of the numerical methods.

Figures (17)

• Figure 1: A schematic description of the SSD: (a) a toroidal thin film on a flat substrate; (b) the cross-section of an axis-symmetric thin film in the cylindrical coordinate system $(r, z)$. Here, $r_i$ and $r_o$ represent the radius of the inner and outer contact lines, respectively.
• Figure 2: The time history of the relative volume loss $\Delta V(t)$ and the energy $E(t)/E(0)$ using the $\mathbf{P}$-method with respect to matrix $\boldsymbol{B}_0(\theta)$, where $h=1/80$, $\Delta t=1/160$, and $\beta=0, 0.05, 0.07$.
• Figure 3: The time history of the relative volume loss $\Delta V(t)$ and the energy $E(t)/E(0)$ using the $\mathbf{P}$-method with respect to matrix $\boldsymbol{B}_1(\theta)$, where $h=1/80$, $\Delta t=1/160$, and $\beta=0, 0.05, 0.2$.
• Figure 4: Time evolution of the mesh ratio $R^h(t)$ for the case of 4-fold $\gamma(\theta)=1+\beta\cos(4\theta)$: (i) weak anisotropy with $\beta=0.05$; (ii) strong anisotropy with $\beta=0.3$. We select $h=1/160$, $\Delta t=1/160$, $\sigma=-0.6$, $\eta=100$ in this test, and adopt the surface energy matrix $\boldsymbol{B}_0(\theta)$.
• Figure 5: Time evolution of the mesh ratio $R^h(t)$ for the case of 3-fold $\gamma(\theta)=1+\beta\cos(3\theta)$: (i) weak anisotropy with $\beta=0.06$; (ii) strong anisotropy with $\beta=0.3$. We select $h=1/80$, $\Delta t=1/80$, $\sigma=0.6$, $\eta=100$ in this test, and adopt the surface energy matrix $\boldsymbol{B}_1(\theta)$.

Theorems & Definitions (14)

• Remark 1
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Theorem 4.1: volume conservation
• proof
• Remark 2
• Theorem 4.2
• proof