Solid-state dewetting of axisymmetric thin film on axisymmetric curved-surface substrates: modeling and simulation

TL;DR

This work develops a sharp-interface model for solid-state dewetting (SSD) of axisymmetric thin films on axisymmetric curved substrates by deriving the governing equations from thermodynamic variation of an anisotropic surface-energy functional. The model reduces to a curve evolution on the substrate and is reformulated with a symmetrized variational framework, enabling unconditionally energy-stable and volume-conserving parametric finite element methods (PFEM). The proposed methods exhibit second-order convergence and robust energy dissipation across isotropic to strongly anisotropic regimes, and enable exploration of phenomena such as particle migration on curved substrates, pinch-off, and edge retraction. These structure-preserving schemes provide reliable numerical tools for predicting SSD morphologies on complex curved geometries with potential applications in micro/nano-device fabrication and materials science.

Abstract

In this work, we consider the solid-state dewetting of an axisymmetric thin film on a curved-surface substrate, with the assumption that the substrate morphology is also axisymmetric. Under the assumptions of axisymmetry, the surface evolution problem on a curved-surface substrate can be reduced to a curve evolution problem on a static curved substrate. Based on the thermodynamic variation of the anisotropic surface energy, we thoroughly derive a sharp-interface model that is governed by anisotropic surface diffusion, along with appropriate boundary conditions. The continuum system satisfies the laws of energy decay and volume conservation, which motivates the design of a structure-preserving numerical algorithm for simulating the mathematical model. By introducing a symmetrized surface energy matrix, we derive a novel symmetrized variational formulation. Then, by carefully discretizing the boundary terms of the variational formulation, we establish an unconditionally energy-stable parametric finite element approximation of the axisymmetric system. By applying an ingenious correction method, we further develop another structure-preserving method that can preserve both the energy stability and volume conservation properties. Finally, we present extensive numerical examples to demonstrate the convergence and structure-preserving properties of our proposed numerical scheme. Additionally, several interesting phenomena are explored, including the migration of 'small' particles on a curved-surface substrate generated by curves with positive or negative curvature, pinch-off events, and edge retraction.

Figures (20)

• Figure 1: Particle on a curved substrate with positive/negetive curvature ZHAO2024120407 (left panel); particle on an axisymmetric curved-surface substrate generated by positive/negetive-curvature curve (right panel) .
• Figure 2: A schematic illustration of SSD: (1) a toroidal thin film on a curved-surface substrate (left panel); (2) the cross-section of an axisymmetric thin film in the cylindrical coordinate system $r, z$ (right panel). $c_i$ and $c_o$ represent the arc lengths of the inner and outer contact points, respectively.
• Figure 3: A schematic illustration of an infinitesimal perturbation (represented by the red line) of the generated curve in the radial direction: toroidal thin film (left panel) and island film (right panel) on a curved-surface substrate, with $\boldsymbol{X}^{(1)} = (r^{(1)}(s), z^{(1)}(s)) = \varphi(s) {\boldsymbol{n}} + \psi(s) {\boldsymbol{\tau}} \in (\text{Lip}[0, L])^2$.
• Figure 4: Illustration of the volume $H(c_1, c_2)$ in \ref{['def H']}
• Figure 5: Three types of initial generated curves/curved substrates.

Theorems & Definitions (13)

• Remark 1
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Lemma 4.1
• proof
• Theorem 4.2
• proof
• Theorem 4.3