A structure-preserving parametric finite element method for solid-state dewetting on curved substrates

TL;DR

The paper develops a structure-preserving parametric finite element framework for a 2D sharp-interface model of solid-state dewetting on curved substrates with anisotropic surface energy. By employing an arclength parameterization of the substrate and a symmetrized weak formulation with a stabilizing matrix ${\mathbf{Z}}_k({\boldsymbol{n}})$, the authors achieve unconditional energy stability and exact mass/area preservation through a discrete normal correction. A nonlinear system is solved via a hybrid Picard/Newton scheme, with numerical results demonstrating robustness, second-order spatial convergence, and faithful reproduction of dewetting patterns across convex/concave substrates and isotropic/anisotropic energies. The approach enables accurate, mesh-friendly simulations without remeshing, providing a practical tool for understanding SSD on complex substrates and guiding experimental design. The key contributions include the arclength-straightening trick for curved substrates, a structure-preserving weak form, discrete area/mass controls, and an efficient hybrid solver for the resulting nonlinear system.

Abstract

We consider a two-dimensional sharp-interface model for solid-state dewetting of thin films with anisotropic surface energies on curved substrates, where the film/vapor interface and substrate surface are represented by an evolving and a static curve, respectively. The model is governed by the anisotropic surface diffusion for the evolving curve, with appropriate boundary conditions at the contact points where the two curves meet. The continuum model obeys an energy decay law and preserves the enclosed area between the two curves. We introduce an arclength parameterization for the substrate curve, which plays a crucial role in a structure-preserving approximation as it straightens the curved substrate and tracks length changes between contact points. Based on this insight, we introduce a symmetrized weak formulation which leads to an unconditional energy stable parametric approximation in terms of the discrete energy. We also provide an error estimate of the enclosed area, which depends on the substrate profile and can be zero in the case of a flat substrate. Furthermore, we introduce a correction to the discrete normals to enable an exact area preservation for general curved substrates. The resulting nonlinear system is efficiently solved using a hybrid iterative algorithm which combines both Picard and Newton's methods. Numerical results are presented to show the robustness and good properties of the introduced method for simulating solid-state dewetting on various curved substrates.

Figures (13)

• Figure 1: A schematic illustration of SSD with anisotropic surface energies on a curved substrate.
• Figure 2: Illustration of the area $F(c_1, c_2)$ in \ref{['eq:area']}.
• Figure 3: Four different initial setups for a initially deposited thin film on convex$\slash$concave substrates with different positions (i): a symmetric film on a concave substrate; (ii) a symmetric film on a convex substrate; (iii) a non-symmetric film on a concave substrate; (iv) a non-symmetric film on a convex substrate.
• Figure 4: Convergence test with different setups : (a) case (i) with isotropy ; (b) case (ii) with isotropy; (c) case (iii) with anisotropy; (d) case (iv) with anisotropy.
• Figure 5: The time history plots of the normalized energy under different time steps (a) case (i) with isotropy; (b) case (ii) with isotropy; (c) case (iii) with anisotropy; (d) case (iv) with anisotropy.

Theorems & Definitions (17)

• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Remark 1
• Remark 2
• Theorem 3.3
• proof
• Theorem 4.1
• proof